The timer on the prompt to accept or start a bounty should be reset after an edit of the question by the OP.
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No, the timer is to force the bounty to eventually be awarded. If edits reset the timer, then the OP could just make an invisible edit to the question every 6 days, and keep the bounty on the question for as long as he wanted, instead of half of it being rewarded to someone after a week.
No, I don't think so. Edits shouldn't change the question. Edits are to improve questions (and answers), fix spelling and grammar, etc.