Does challenge progress carry over -- not just the stat boosts rewarded by them -- between characters on one account? For example, if I'm working to kill 1,000 bandits, can I achieve that with 500 on one character and 500 on another?
No, each character has to complete the challenge themselves.
This is a good thing, however -- because you get an additional badass rank for each completion. Meaning more Badass Tokens.