Why wasn’t Edmure married with and/or trying for kids (besides needing to be single to be put out to stud for the Red Wedding)? Seems like it should have been an important move for House Tully to make sure they have heirs aplenty (though there were a bunch of spare Starks). And since he wasn’t, wouldn’t it have made sense for Bran (as heir in the hole) to have been foster at Riverrun?

It’s an interesting question. I can think of a number of possibilities: first, it may be that the fallout from the Blackfish might have made things more difficult if Hoster’s ability to speak for his kin was called into question. Second, the damage to his reputation by Tom o’Seven’s song might have made it more difficult – no one wants to marry their daughter to a man who potentially can’t consummate the marriage and sire some children. Third, given Edmure’s love for wenching, it may well be that he didn’t want to get tied down too early, so he may have resisted or drawn things out. Fourth, given that the Freys seem to have been pursuing him quite fiercely – it may be that Hoster was holding off, lest he offend some of his vassals by picking one and not another. 

As for Bran, I don’t think Eddard would have allowed that. He seems to have been very anti-fostering, probably due to the all-but-total loss of his own family at such a young age.