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.
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For some of the more annoying or time-consuming challenges this can be a pain, but there are plenty you will achieve just by playing through the game naturally (# kills, finding guns of X rarity, amassing fortune, etc) – Alex Aug 14 '13 at 19:45
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Plus, it's not actually possible to fail one as there aren't any time limits. – Shadur Aug 14 '13 at 19:56