Do the champs usually just get trophies and bragging rights, or are there also monetary rewards to a lot of the national or world martial arts championships?
First off, there are no real national or world martial arts championships. It's not like there is some World Martial Arts Association that regulates such things. Some specific, well-organized arts might have such things, but not for Martial Arts in general. A lot of different tournaments call themselves that for marketing reasons. This is why you should never trust anyone who tells you they were a national or world champion.
End of that rant.
Yes, some tournaments have cash prizes. Most have trophies, huge overbearing trophies in many cases. As for bragging rights...a good martial artist doesn't brag. He doesn't have to.