Why couldn't Cora become queen?

That's a mystery that will probably be revealed at a later time. Just because she's dead does not mean they won't be bringing her back in flashbacks.

I assumed that the King had multiple sons, and Cora married one of the younger ones. So if he had four older brothers, that would make him (and her) fifth in line for the throne.

I suppose there was something wrong with the entire kingdome. The prince should have been Henry, since the baby born was Regina, but Henry is no king. Is Henry not Regina's father, or did something happen to that royal family?