It's pretty easy to have a genetic structure shared across 3 sexes and combine across the three. You could have either one lay egg or one carry after the three sexes combine chromosomes.
However, justifying the evolution of this species is harder. We have two sexes because we need a way to trade genetics; however, once you have some way to recombine genetics there is limited use for a third sex. If two of the sexes reproduce then each of their children has half their genetics if three reproduce then each child has 1/3 of a given parent's genetics. Once you have some means to add genetic variation the extra genetic variation of a third parent does not add much more to the overall genetic health of the child compared to two parents, and the loss of your child not carrying 1/6 if your genetics it otherwise would have is significant.
There is a strong evolutionary advantage to evolve to only need two of the parents if a three-parent system existed; more importantly, it's very hard to imagine a situation where a third parent would evolve in such a situation like this. It's very hard for two parents to evolve, but it's almost certain that a two parent system would develop before a three-parent system would, and once a two parent system exists there is little reason for 3 parent system to evolve.
This is relevant because I can not answer most of your questions until I know how the evolution occurred. If I can pick from one of a number of evolutionary options I can only then describe a plausible explication for genitalia, culture, and physiology of the species in question. Thus the first question, to create a believable world, is how could this evolve. Let's see our options:
Have a hive with a queen mating with males and many drones that never get to mate. In this situation, it's possible for a gender to evolve that does not benefit from spreading its own genetics, but from supporting it's 'queen' from spreading her genetics. In a situation like this one parent may share little or none of its genetics directly with the child; but it's a child, or close genetic relative, of one or both of the other parents and thus gains from helping them.
Imagine a female queen that has male drones to mate with and a 'brood' sex that carries the child produced from the male and female mating to term, but does not contribute any genetics herself, being just a uterus for carrying children to term. The brood sex will be a close genetic relative to the queen, either daughter or sister, and thus shares 50% (or possible up to 66%) of her genetics with the queen. By helping her queen produce lots of children the brood sex benefits from spreading her queen's genetics even though she herself does not spread her own.
The female may produce lots of these brood-sex children, possible many dozens. As each of these children births, one match of young the queen will summon her up to mate again with the queen and her male, to produce a new batch of children. Eventually when one of the brood sex dies the queen will choose to produce another of this sex to replace the lost one.
The big benefit of this approach would be that the queen can produce more children at once. With a traditional queen situation, a queen needs to produce lots of eggs, which requires a massive caloric intake and a huge body for processing these calories and translating them into young. With a brood sex the queen can force the brooder to put in all the caloric investment to carrying her young, and she can have dozens producing young. This makes it easier to provide nutrition to each one-the queen is spatially limited in her ability to have kids because she can only convert honey to eggs so fast, but she can always add more broods. This also means the queen can stay small, and thus mobile, instead of being a huge thing in the center of a hive that can't move. Perhaps a queen can even have multiple hives that she flies to long enough to impregnate each of her brooders, before flying back to the first hive (though this would make her vulnerable to predation; this could still work of new queens are always ready to replace the first when she doesn't come back and may add to the genetic diversity of the hive if it's forced to replace its queens on occasion?
The downside is that each of these brood children requires extra calories since multiple 'brooders' is not as efficient as one super adapted queen producing all the young. Thus this would only work in situations where producing lots of young quickly was more important than conserving resources, and it's hard to imagine this being the case often. Perhaps if the hives were mobile, one hive would set up and consume the resources in an area quickly, then break up and fly away to set up more hives. The ability for a queen to start a hive full of brooders while she travels to set up a new hive may be useful.
There are a few ways to handle sex in this format, depending on how the queen and the brooding sex differ (the standard male role would be pretty much as it already exists).
The queen may serve a role closer to male and simply providing more genetics to the brooding sex, being unable to produce any children without a brooder. in which case the queen would need to be provided brood sex 'sisters' from her mother when she starts a new hive; replacing these brooders with her own daughters as they die out. This seems least likely since it means the queen and hive are worthless without brooders.
Alternatively, the queen may be able to produce only brood sex children herself, but nothing else without a brooder. I could only see this if a queen could produce both brood sex and males, and is using a system where she can start a hive by herself by birthing the original ones in a Haplodiploidy system, but needs both sexes to produce new queens.
Most plausible, the queen could carry any child to term herself but prefers to use brood sex whenever possible (possible in conjunction with Haplodiploidy system above). The queen's first mating may have her birthing some males and brood sex children herself, then from that point her she uses her brood sex children for all matings.
The reason I consider the third option to be the most possible is that it's the most likely to evolve. As I said a 3 sex system is hard to justify, but in this case, the 3 sex system is an obvious offshoot of the traditional 2 sex system. The brood sex may be genetically very close to queens, except sterile by themselves. They evolved as queens that were intentionally kept sterile by their mother and exploited, eventually evolution causing them to diverge as more unique sex. The queens retain the ability to birth young (be it egg or live birth) the same way the brood does, but the brood traded ability to generate eggs for an ability to carry more eggs to term etc.
To answer your questions in this case the brood would likely be referred to either as it or with female pronouns like the queen. Since the brooders would not be given much attention, and likely wouldn't be intelligent, it will not be confusing to use the same pronouns as you use for a queen, any more than calling both your wife and your dog 'she' would be confusing.
Both queen and brood sex would have very similar genitalia. I imagine the females (both sexes) would have an ovipositor that can be inverted to act as an oviduct. TO give a rough analogy something like a 'vagina' that can be pushed out to act as a 'penis' for ejecting eggs into the broodmare. Both the queen and broodmare would have very similar structures, even if the broodmare never ejects eggs into others.
Culturally the broodmare doesn't count as anything of importance. You have females and males, and biological things that their child is growing in.
Shared parental investment
Have three independent sexes, which all invest equal effort into raising the child. It's important that they all contribute equal effort, if they do not it becomes hard to explain how this is an evolutionary advantage, since one sex gets to spread their genetics with less expense; which works in our world only because the advantage of sexual intercourse is so high that females are willing to pay a disproportionate expense to have sexual reproduction; once you already have at least two sexes reproducing sexually there is no reason for allowing a third sex to contribute less effort but get the seam genetic benefit; that would be evolved away on the grounds that you can do two-person sex without the free-loader.
There are many ways to get an equal genetic contribution, though most approaches favor R select species which are unlikely to evolve sapience. Here are the key ones I can think of.
With R species all three work together to produce a large batch of eggs somehow. The energy investment for each sex is about equal in producing the eggs. and the eggs are then left alone to produce lots of young without further parental investment.
Close to one, each sex takes turns in raising the young through a phase. sex X produces the original eggs which it transfers to sex Y, likely through some form of internal fertilization, at a minimum to an egg pouch similar to what seahorses have. sex Y fertilizes the raw eggs and carries them partially to term, long enough for the eggs to develop a hard shell and be safely 'layable', ie able to survive outside of the body of X or Y. Y then lays the eggs so that Z can fertilize them. Z then protects the laid eggs from predation, sitting on them, protecting the clutch etc, until the eggs hatch. At this point, the children are mostly forced to fend for themselves.
In a more K select species, one sex produces large, caloric expensive eggs, that it passes off to the other two sexes. The other two sexes fertilize the egg and then work together to raise the young. The put little caloric investment in going from embryo to born child, but the invest the effort in raising that child to reproductive success. Though this seems least likely, it's the one most likely to create sapient species.
All species 'conceive' after intercourse. Perhaps three 'eggs' (very underdeveloped) are produced from intercourse and each sex will take and care for one of the produced young, with the primary investment being parental care raising the egg to adulthood, not in the production of the egg itself. Why you would have 3 sexes instead of one (see below) in this situation is hard for me to explain though. In terms of intercourse, I would probably do something like what I had in 2 though. Sex X and Y have internal fertilization/intercourse to produce 4 large 'layable' eggs, X and Y both take two of these eggs. they both find a sex Z and have intercourse with it to fertilize their two eggs again, and both sexes walk away with one fully fertilized egg to care for.
Your notice in all these situations I suggest that the sexes have intercourse at separate times, and that sex X may never know the sex Z that fertilized their egg, this is intentional. It seems quite difficult to justify an evolution where 3 sexes have to agree on each other as a mate, what if the X doesn't like the Z but likes the Y? how do per-sapient sexes handle mate selection and decision making? It's more plausible if each sex needs only pick a single mate at a time to have intercourse with, even if another's DNA will be added later. This also justifies the only evolutionary advantage I can think of for 3 sexes if there is a reason to delay the development of the full child. Perhaps environmental factors can cause embryos to fail to form early, so having a phase of partial-development (after X and Y mate) allows one to see if the children will successfully develop or die due to undesirable mutation/chromosomal damage, prior to final fertilization and the completion of a child developing.
It's possible that in sapient species sex X Y and Z will all know each other, but the original mating habits likely evolved around X never knowing Z, and it's only with later sapience that interest in knowing the full genetic investment in the child lead to X and Z meeting.
In terms of pronouns, I think you would simply have to create new ones for almost all of these, such as the way I said sex X Y and Z.
In terms of genitalia, I would imagine all have very similar genitalia, with only minor differences. They should all look similar since they have to all evolve from a single format. I can't say more without knowing which of the above options were used.
These systems would likely lead to R strategies where parental investment didn't exist, and thus no sapience. For the ones that lead to parental investment originally I would imagine each would raise their own child, but eventually, cultures may move towards all three sexes living as a family unit.
Hermaphrodite single-sex species
Finally, there is a third option which I consider far more likely from an evolutionary standpoint, which may still meet your needs. Have only one sex!
Instead of multiple sexes, which as I said are really hard for me to see evolving, have one sex, but which is capable of mating with multiple individuals of the same sex and producing a single child that shares DNA with three, or more, parents
With a system like this, you could claim that DNA can be split and recombined safely in more then than just binary manner. Perhaps each chromosome can be received from a different mate, or even individual chromosomes are broken into smaller gene sequences which can be split and recombined to combine genetics of multiple mates. This is not an impossible system, I think even our current DNA could evolve to an approach like this without too much hassle (potentially a minutely higher chance of birth defects, but the genetic fitness of multiple mates could more than make up for that), and a system that encodes genetics differently then DNA could definitely evolve for an alien species where this makes sense.
There is a genetic advantage to having your child contain DNA from multiple parents, the more genetically diverse a child is the more 'fit' he is on average, but the advantage is only so large. For the mother to benefit she must make sure she is still ensuring that half of her DNA is inherited from each mating she performed, otherwise she loses from the mating. There are a few options for handling this:
Your child always caries half your DNA. When mating with others you recombine their DNA to form the other 50% of your child's DNA. Since the mother always provides half the DNA it never hurts her to mate with more mates, and she actually gains more genetic diversity the more she mates with others. In addition any mating the mother has allows her to provide DNA to the female she is mating with, potentially contributing some DNA to her mate's child as well. There would thus be major motivation for females to mate with multiple partners, expect a very 'free love' society.
While it's possible to limit the mother to being able to only mate with 2 mates per child, with a system like this it feels more natural to allow many mates to all contribute DNA to the child, Once the mother has the ability to recombine DNA from multiple 'fathers' why should she stop at two?
However, since each 'father' may very well only contribute as little as 1/10 or less of DNA to a resulting child the father would not be inclined to provide much, if any, parental investment into the children of a mother. Thus this race would likely consist of all single parents.
Insects already have a variant of this, in which they produce many young with each young having potentially different fathers, so look at them for mating adaptations. Most noticeable it's likely that each mother will have some degree of control over how much, or if any, of a mate's DNA contributes to that of the child. This control may be conscious or subconscious. It may be that a mother attraction or pleasure with a mate will result in that mate's DNA being a larger percentage of the child's DNA; but that they mother has not direct conscious control over this factor. How much control the mother has in the percentage of DNA shared will effect mating behaviors.
In systems like this sexual mating may be traded socially. Look at the bonobo, who use sex as a social feature. A mother may mate with lots of others to please them. If she has strong control over who's DNA is used for the child she may mate with everyone to obscure her mate choice. If she has less control (ie any mate will likely contribute at least a little DNA to the child) she may be more selective with mates, but may be willing to mate socially with someone who has earned the right, through gifts or favors. In any case since any father only contributes so much DNA it's safe for her to be less selective about mates, the less control and the more DNA the child will potentially posses from any given mate the more selective she will be.
Pronoun wise you would have mother and daughter, and that's it. You may use 'mate' to refer to those that mated with your mother, but since the child of a mother may have little if any DNA of a given mate the concept of father doesn't really mean much. sisters will exist, but only if they share a mother.
Genitalia wise you have your standard penis and vagina. Or possibly an organ that combines the two together to allow simultaneous transfer of sperm to both sides.
Mother only provides 1/(mates + 1) dna to the child (limit to only 2 mates)
This gets back to the situation where it's less advantageous for the mother to mate. If the first mate produces a child with 1/2 of your DNA, and mating again produces a child with 1/3 of your DNA then there is little incentive to mate a second time and have a child with less DNA, usually. One easy way to fix this is to force mating to be mutual. If a mother mates with another female they are both guaranteed to conceive a child that possesses the mates DNA.
This works because each subsequent mating allows the mother to spread more of her genetics to the next generation, even if the child she is carrying possesses less of her DNA. Say after the first mating she will conceive a child with 1/2 of her DNA, and she chooses to mate with another mate. Her child will now possess only 1/3 of her DNA, however, her mate's child will possess wither 1/2, or 1/3 (if her mate has mated with another) of her DNA as well, meaning that The subsequent mating has resulted in her child having 1/6 less of her DNA, but another mate's child having 1/3 of her DNA; that's a clear benefit to the mother.
However, if the mother can mate with an infinite number of other mates she reaches an odd point, where she doesn't care about her own child. Once she has mated with 9 mates her child only possesses 1/10 of her DNA, and thus is barely close enough to warrant caring at all; she is more interested in the other 9 children who also carry 1/10 (or so) of her DNA. This makes for a rather odd, and eventually unstable, mating solution.
The easy way to fix this is to limit the number of mates a mother can have, in fact a system like this would almost have to place a limit on total number of mates; otherwise matings become so common that there is little intensive to care for your own child and things would evolve to an R select strategy of having dozens of young and tossing them into the world to fend for themselves.
So, say the mothers can only mate with a total of 2 other mates, this is as much as DNA can viably combine. Now each mother benefits from having two mates. IF the mother only mates once she will have one child with 1/2 her DNA and one with 1/3 (most often, if her mate has a second mate) or at most 1/2 her dna. A mother who mates twice passes on either 100% or 5/6 of her DNA to the next generation, depending on her mates mating habits. A mother who mates with three mates has at least 3 children that all cary 1/3 of her DNA, so she at minimum passes on 100% of her DNA, and there is always a small chance of passing on more if one of her mates doesn't find a second mate.
In addition the 2 mate option means more genetic diversity over one mate, meaning 2 mates is still beneficial over mating with a single mate who will not mate with anyone else. Plus, since you can never be sure your mate won't try to mate with someone else later making sure you mate twice means you come out ahead no matter what your other mate does.
This system is, again, plausible to evolve. Though a 3 mate system (meaning that DNA is split into 4, which is more likely due to it's being divisible by 2) may be more likely then a 2 mate system. In this system a mother may be a single mother, or one or both of the mates may stay to raise their children as a family. Occasionally two mates may choose to have a child together without involving a third mate as well, far more common once the species reaches sapience.
This system would lead to mothers being slightly less sexually selective then females in most mating situations of today, but still nearly as selective.
For this system to work there must be a pretty strong promise of conception occurring with all mates, which adds a potential complication. Females would have to be able to conceive very soon after mating, with very clearly defined mating 'seasons' during which they will conceive if mated. More importantly, and difficult to evolve, a female would need a way to advertise once she has conceived, say by swelling going down, as soon as she accepted her third mate. If this didn't occur then a female could mate with other females after accepting her third mate and impregnate them, spreading her genetics, while effectively ignoring their DNA offered to her since she already conceived.
There would be very strong evolutionary advantage for a female to evolve a way to avoid conception from a mating, or simply to mate after she has conceived but is not yet showing. To 'trick' other females into agreeing to carry a child with part of your DNA while you don't accept the need to carry any of their DNA allows you to mate with many females, and potentially spread far more of your genetics. This is a bit of an issue with this option, making it less likely to evolve; or more accurately to be evolved away from not too long after it's original evolution as people start finding ways to 'lie' about their mating.
This could be fixed with a monogomous mating system, where mates all know and watch each other. Thus a mother may choose to mate with a second female only if she can watch the female and verify the female does not attempt to mate with more then two mates. If a female does manage to sneak a third mating she will benefit from spreading her genetics to more, at the expense of her first two mates (who may share less or none of their genetics if she mates with a third mate).
This monogomy could take two forms. One is which all three mates choose each other and mate year after year. As I said above this is harder to manage in non-sapient species, but not impossible; and it's more justifiable when everyone is the same sex; making it easier to choose any other female as your third party mate. In this case the females would likely mate between themselves and raise their young as a third person family.
Another option would be that a female will stay close enough to watch her mate, but not try to control her mates selection of mates. Effectively female A will declare she mates with B to everyone. When female C comes by B may choose to mate with C, which A will allow; but if B ever tries to mate with a third female both A and C will protest since they are aware of her two chosen mates (A would have advertised to C that she already mated with B). meanwhile A and C may choose mates of their own. B will watch both of them and ensure that A and C only choose one other mate, but does not try to prevent or control who that third mate these two choose is. This mating strategy works best when everyone mates in a nesting area (or village) where it's easy to watch your other mates. Perhaps A B and C will nest closely together and each will contribute to helping raise B child partially, while also focusing partially on raising their own and their second mate's child. This sounds complex, but I think it could evolve to work pretty well. Each female has incentive to stay near both her mates, because both mates contribute partially to raising of her young, so it's easy for those mates to watch her to ensure she doesn't mate again.
Genitalia wise you would likely have some system that allows sharing of sperm by both mates at the exact same time. I would guess some sort of 'dual plug' system, effectively the same as inserting your penis into her vagina the same time she does the same to you. The positioning of the 'plugs' would have to make mating easy though. Mating from in front would be preferable then behind (behind makes more sense when the male and female roles are separate, due to female having more of a need to 'defend' against unwanted matings then male). I imagine their organs would simply be more flexible to allow repositioning to align up with someone your facing.
Pronouns work as above, everyone uses female pronoun but mother is limited to the female carrying the child, while those females that mated with the mother are considered father's of the child the mother carries. Yes that means a female can be a mother and a father. Siblings would get complex terminology. you can have third and 2/3 siblings (probably a more convenient way of saying 2/3 sibling will evolve). You may have two words for sibling, one for "we have the same mother " and one for "we don't have same mother, but share partial DNA" as well. Thus you can have third, partial, and full Sisters (share mother), and third, partial, and full siblings (different mother).
In a system like this some degree of parental investment will occur from all mates for each child, since it's almost as useful to care for your partner's children as your own since theoretically all three Cary 1/3 of your children (though not quite as much, since you can never be 100% certain cuckoldry didn't occur with partner children).
Culturally I imagine there will be both 3 parent homes where the parents only mate with each other and raise children together, single parent homes where a parent raises their child alone without aid or investment from the other parents (less common), and shared communal homes where people work together to help raise children.
The communal homes are most interesting. Take my example above where A and C mate with B. Now imagine A also mates with D, and C mates with E, and for the sake of simplicity C and E happened to mate. Now you have a group of 5 parents that all share children with two other parents. The parents would all prefer to stay close to their other two mates, which could result in all 5 mates living together.
Now imagine if it got more complex, you could have 10 or 20 mates involved in a chain before the chain of matings 'loop back' by having someone mate with two people already in this mating chain. You may tend towards large communal living spaces where everyone tends to contribute and share parental care of children, but of course they all mostly focus on the children the sired. I think the entire species would evolve to have more of a 'community raising the child' mindset due to this bizarre spreading of genetics. How much this is true depends on how monogamous the species is.
Speaking of which, this species could be highly monogamous, or each mating season it could mate with different couples. I would imagine it would tend more towards monogamy, as it would make it easier to share parental investment of raising multiple generations of children at once, but it need not be mandatory.
This species will most likely have mating seasons or some similar concept, everyone becomes fertile at roughly the same time. That's required to make watching your mate and ensuring they only mate with one other mate possible.