This is what it boils down to. Also there are career issues to consider. A great deal of responsibility for the real world differences in income between the two genders has a lot to do with the demands reproduction makes against a woman's career. Studies have shown that women tend to go for lower-paying fields that are more forgiving of time off for family and child-related duties, and are less likely to remain in high-pressure, demanding jobs for long periods of time. In addition, many careers require you to maintain an up-to-date knowledge of the field. A year off for maternity leave for a software engineer can leave one hopelessly out of date. While this might not seem to be true for channelers, an Aes Sedai's job is not really channeling, but being Aes Sedai. That requires either involvement in Tower politics or being on the road doing things. The latter is absolutely counter-indicated for pregnant women or those with young children in a world with no motor transportation. As for the Tower politics, that involves staying up to speed on events and schemes and plots and whatnot. Children are a vulnerability in that kind of thing, pregnancy appears to be some sort of weakness in the eyes of other women, however hard they close ranks on any male who dares belittle that institution, and the importance they attach to Tower business suggests they cannot see stepping out of the loop for a couple of months for what they have been "raised" to see as a frivolous pasttime.

I think sisters exaggerate that last - men are notoriously willing to go along with a sexual relationship no matter what the glaringly obvious unsuitability of the partner in question. Meanwhile, women are notoriously more desirous of long-term relationships, but any relationship with a non-channeler is, from the perspective of a channeler, automatically a short-term thing. Going by sexual generalization, it's the women who are far more to blame for the lack of relationships in this case. And why on earth does the lack of relationships preclude having children? While it's not optimal for a child to be raised solely by a woman, an Aes Sedai is in a position to mitigate most of the worst aspects of that situation. Anyway, it's not like she'll be doing most, if any, of the work of raising the kids, so she can arrange for male supervision or care as well.

The impression I got was that marriage is discouraged, not expressly forbidden, since the only Blues we really know both have hard and fast plans to marry. "abstain" was the word used in New Spring, though I don't recall if that is merely a custom or an express requirement.

I think it's just an excuse, since that does not appear to detract from relationships among the Sea Folk or Aiel.

From what Moiraine said, that was through the male line. The Kings in that case would simply allow the woman they wanted as queen to bond them. Even if it was true that the king & queen were always warder & Aes Sedai, it does not follow that they would have to be married, anymore than the King and Panarch of Tarabon have to be married, or a clan chief and his main Wise One advisor, or a Sailmistress and her Cargomaster, or for that matter, than a Queen of Andor & her official Aes Sedai advisor have to dyke it out. The actual notion that the queen was always an AS who bonded the king is no more than a rumor Faile stated and strongly suggested to be unfounded.

There is no suggestion of any obstacle illegitimacy presents to succesion anywhere in WoT, and given the primacy of female power, there is also no reason for such a cultural more. There has not been one suggestion, in the three successive books in which Elayne was pregnant that her unmarried status carries the least implication for her reputation or the succession of her children. Of greater importance on that score in Elayne's mind is not having twin girls, than in arranging a marriage to legitimize her bastards. A crisis of succession inspired by an arguable seniority of claims is a greater threat to a smooth succession than any taint of illegitimacy undermining their claim.
In a case where succession comes from the female line, there is no reason for illegitimacy, since all children are unquestionably the offspring of their mother and there is no need for legal protection of a woman's rights when she is the one with the power and resources.