I keep winning fights, but my honor only goes down. How do I gain more honor? And what happens if my honor gets too low?

You'll need to attack players that it says will give you positive honor (The amount is listed on the attack button).

Honor can be gained by attacking a player that has positive amounts of it. The value given by other players depends hugely on your fight balance; the more imbalanced you are, the more it leans towards positive or negative. If your honor dips below a certain threshold, you will not be able to attack another player, so watch your honor level specially if it gets too low. Also, if you attack a player with negative amounts of it, well, you'll lose your honor based on the exact amount that player gives you.