Adding another answer because of a very different idea for approaching what you want. This is one I have toyed with before, though it would be one that is hard to implement well; still I think it would cover all your desires and be quite interesting.
Imagine a species where their genetic coding is somehow more module, making it possible to use the DNA (or equivlent) of multuple 'parents' at once. Perhaps the 'mother' could collect DNA from multuple fathers and merge then to create one child, for instance perhaps they could use a chromosome of one father and a few chromosomes from another and most from a third. Maybe the amount of DNA used from each potential father is decided by how much the mother likes or respects that father, the more 'attractive' the father is the more of that species DNA is used in the resulting child(ren). Now imagine a hermaphrodite species where each organism is capable of providing said DNA, and of carying a child.
In this world the collection of DNA would be little more then acknowledging that you feel someone has positive traits you would like the child to carry. Since the mother may use DNA from many 'fathers' the concept of paternity may become murky, to the point that no one bothers to ask who the father is if he is only 1/15 the father, and the father(s) would likely play no significant role in the child's development. This would drastically devalue sex from both ends, since each sexual encounter is less relevant. Perhaps sex would be a social activity, someone who respects another would touch his organ to another and dna would be exchanged quickly and easily, perhaps in public; it could even be as simple as a handshake. Eventually, when someone decides to have a child they could make a conscious decision to merge the DNA they have collected to produce a child.
This would devalue sex and remove gender entirely. In essence there would only be a concept of 'mother', since the mother was half the DNA and everyone else a tiny fraction. The potential social results of this vary greatly depending on how you develop it. Some examples
1) the mother has no control over dna merging, anyone she has 'sex' with is equally likely to be part of the child's DNA. In this version sex would still be limited only to those that are respected, and thus have some cultural importance. However, sex would likely be fast and not nearly as taboo or major as it is today. Perhaps it's a common token of respect done between two people, but still done semi-regularly. An important gesture, but not life changing.
as a perfect example of the culture of a species like this may evolve look at the bonobo, my favorite primate. Only one male will ever father a child, but the dynamic of sex is very different. In bonobo land since females mate with SO MANY males no male view sex as a gaurentee of paternity, instead sex is a social gesture. Sex leading to a small percentage of your DNA being used is very similar to sex bonobo's world where sex leads to a very small percentage chance of all your DNA being used. Look at every instance where bonobo use sex as a social function and mirror them in your species and you will have a pretty realistic culture.
2) Say the mother can engage in sex, but can choose to discard the DNA of someone they don't actually like. Then sex would be a more ritualized act. Perhaps it would be as simple as a handshake, though I think it would be considered slightly more relevant. Perhaps something closer to a kiss/hug, and a salute or sign of respect. something you do to show you like or respect someone, but still done regularly. the standard military salute may very well be sex, since it's a sign of respect but also anyone who does not desire the saluted person can simply discard the dna from the exchange so it is safe to do it for cultural reasons without it effecting offspring. Keep in mind at this point 'sex' would have to be something easily done as simply as we kiss, which is pretty likely evolutionary.
3) Imagine the degree of dna used from each donor is effected by the mothers feelings towards the donor, though perhaps not consciously. The more respect, trust, love, or admiration the mother has the more of dna used from that donor. The level of control the mother has can vary
4) if the mother has full control over the amount of DNA used this could have interesting cultural ramifications. Perhaps it's possible to give birth to a full-clone, but it's culturally frowned upon due to the lack of dna exchange (there are many reasons exchanging DNA is important). Perhaps a sign of deep and strong respect between two people is to use a full 50% of DNA from that person to produce the child, and this situations result in more traditional nuclear families but are less common. Maybe using only one donor to produce a child, and living with them, is viewed much like our culture views homosexuality, different and some think is wrong, while others see no issue with it. There would be encouragement, on an evolutionary level, to use DNA from multiple fathers to increase diversity so using predominately one persons DNA would be the exception and treated as odd rather then the norm.
Getting the science to work here is somewhat difficult, but not impossible. The key thing is to have some concept of chromosomes, an independent 'unit' of DNA which can be used as a building block and combined with other 'units' from other donors. IF DNA could be made module then the above sort of organism would almost certainly evolve, there would be strong evolutionary advantage to it. In fact look at any species that tends towards producing lots of young and your see the mothers trying to have as many fathers as possible for genetic diversity for exactly this reason.