1. What happens if I found a family seat with Clerics (the family that give a religion upon founding) but all religions have already been founded? An unlikely situation to be sure, but I think it could in theory happen?
You will just lose out on the bonus - so nothing special will happen after founding.
2. Is there a way to tell units to heal until fully healed, or must the player click 'heal' every turn?
You have to keep clicking as it stands now.
3. Say I am Rome and I marry a Carthaginian. My child will be 50% Carthaginian. If they then marry another Carthaginian, their child will be 75% and so on. Does this have any effect other than the "heir/leader is Carthaginian" bonus relation?
It plays into opinion in various ways. There are the positive heir/leader/spouse bonuses, but also those part-Carthaginian characters will dislike you if you go to war with Carthage. Then there are some events that trigger if you have a foreign-ethnicity character, but that goes for a lot of things in OW, that they trigger events.
4. Related to the above, is it ever worth marrying into a tribe? It always seems better to marry into a family for the relation bonus or a foreign power to secure a border.
I consider tribes to be the less useful of the three marriage options, but sometimes it's worth it. At higher difficulties, you can definitely end up being very pressured by a tribe, and marriage can be a good way to secure peace. If you want an alliance with a tribe, you need to make them happy, so marriage is an integral part of that.