This question is extremely open ended, and I could think of many possible societies with this biological setting. It's probably safe to assume that many cultures with different views regarding sex and gender would evolve throughout the world in question.
I'm going to name at least one possible structure for a society that has this kind of biological mechanism, and meets your goals of a patriarchy.
If we assume that males are already held in higher regards, we can probably assume that when a child begins to develop male sex factors, there would be some amount of celebration. The only real difference between earth societies would be that you have to wait more years to find out if the child is male or female.
This can lead to two possible scenarios: Children are only birthed every 12-13 years, depending on if you're current child becomes male or not. Or more likely, many children are had to ensure a male heir. The second is more likely since many births were common in medieval times because of high infant and childhood mortality rates.
Let me start by going through your list, and then adding other developments I think likely in personal opinion.
Names: if I had to guess, I would think that most parents would choose masculine names for their children, in hopes that their child would become male. In the case of names with a feminine form, the feminine form could be added if the child becomes female. IE A Samuel/Sam would become Samantha.
In the case of Royalty or nobles, they may choose to bestow male names only to male heirs, an early sign of their right to inherit land or prestige. However, nobles often subscribe to some of the same superstitions as commoners, so they may also practice male naming.
Heredity: Your guess that Royalty would sire many heirs in order to ensure a male scion is most likely correct. It has been observed throughout human history that the more wealth a person has, the more likely they are to participate in Polygamous practices. Royalty would probably have many wives to ensure a proper heir was born.
The selection for the most proper male heir could happen in many ways, but the most likely answer is simply that the first Child to develop male characteristics would be the true heir, and the appointment would pass to the next child who had developed the traits in the event of their death.
In this case a younger child may become the true heir if an older sibling fails to develop sex characteristics first. This would only hold true if the development of sex had a distinguishing factor, such as developing a proto-penis, or having breasts form. If the change is more varied such as our puberty, birthrights might have to be decided by a contest instead, with similarly aged heirs competing for the birthrights. Perhaps they would have all heirs in the same calendar year compete in a contest when they come of age.
Education: Most likely many skills would become more unisex, with hunting and crafting being at first learned by both, and specializations going on after they hit their sexing period. This would mean that men and women are a lot more conscious of the sex split, despite the fact that children are treated more evenly. For children who learn together with their siblings and then suddenly get divided by sex, there would probably be a large emotional attachment to the event.
As for higher education, only nobles would get it in the middle ages, and they wouldn't start until they came of age at 12-13 anyways. Schooling would be another disjoint between the sexes, and a feminist movement may happen a lot sooner than typically expected because as children both sexes had the same rights, and women will want the same rights in adult hood as well.
But until that happens, while noble or wealthy boys went to school, girls would suddenly be given wife lessons in an attempt to marry them off to a man in their mid teens.
The split would be less apparent in commoners where the only major difference would be that girls would start spending more time learning craft, while boys spent more time learning field care and other outdoor activities. The main reason for this is that the wife will need to stay at home while pregnant, and crafts can be done inside. A boys knowledge of crafts won't go to waste of course, as he'll be able to use it during winter, storms, and other events that may keep him inside.
Gender neutral/Intersex: Gender neutral people would probably be regarded as a third inferior sex. It's even possible they would be considered cursed by god and may be scorned or even killed. Some cultures believed that third sexed people were special and were given praise, but it's far less common then condemning outcasts.
Sexing: I can only imagine the wonderful plethora of hogwash sexing methods people would come up with. There would be debates over what sex position gives the most boys, over what kinds of foods children should eat, probably weird practices like feeding a child the placenta from a newborn to help them grow into men.
There would be Shamans who would sex your baby based on earlobes or birthmarks, but it would obviously be bull. Maybe somewhere a scientist would suggest looking for ovums by cutting open a child's belly to sex them, but considering the risk of infection and the great chance that the doctor would be called a heretic for desecrating bodies or something, I doubt any reliable method would crop up.
Patriarchal society: There's actually one very good reason that a patriarchal society would develop even though children don't have sexes, and that is the fact that men don't get pregnant. Yes, in early cultures women were life givers and served as matriarchs, but as time went on men, who tend to have more aggressiveness, would be able to go out and conquer or whatever while their wives are pregnant. Actually, just the fact that men are more likely to fight could lead to power shifting to men. With men on the top, a patriarchal society develops as a side effect. IE: All these men are ruling countries, men must be great and special.
Extra notes: I think that covers a lot of the basics, but I do thing one thing is important to point out. In this type of world, I'm fairly sure the gender gap would be wider, not narrower. With men and women growing up separately from birth, there is underlying complexes, but the comparability between men and women shows them that they are more similar than not. But in a world where the sexes are indistinguishable until a certain point, and then suddenly they are divided into men and women, it would leave both inferiority and superiority complexes on respective parties, which are then reinforced strictly since there would be lingering feelings of "we're the same" from childhood, which need to be gotten rid of if the patriarchy is to continue on and society as well. (Or at least that's how they'll justify treating men like heros and women like objects.)
(I know I mentioned this above, but it seemed important enough to make it its own point.)