How long should I wait before posting a bounty? Two weeks? A month? This is considering the question is cycled upwards. Assuming the question has either no answers or just a few plausible but yet unhelpful answers, and I want to attract more people to answer (seeing as how I have not accepted an answer) how long should I wait for someone else to respond to the question? I would imagine a minimum of a week and I'm only willing to extend past two months. (This isn't meant to be opinion-based. I want the actual experience of users who have posted bounties before or let a question like this simmer for a month.)
It's a judgement call and depends on the specific circumstances. Here are some factors to consider:
The software allows you to post a bounty starting when the question is two days old. (And it must be open.)
Bounties increase exposure, so if your question is currently getting a lot of attention already, the bounty won't make as much of a difference and you might be better off waiting.
If the question is attracting close votes or comments requesting clarification, that might indicate a problem with the question that you should fix before drawing more attention. You want your question to be in good shape before adding a bounty that shouts "hey, look at me! answer please!". A bounty pays for advertising, not for an answer. If the question is unclear or too broad, you might spend those points and still not get an answer.
If you urgently need an answer, you might want to offer the bounty sooner rather than later.
I don't know if there is data about timing. A bounty runs for up to a week, so it shouldn't matter when you start it, but there's a little extra impact of that blue "+100" (or whatever) being on the front page in addition to the bounty tab, so maybe Sunday morning UTC isn't the best time. (Traffic is lower on weekends than weekdays.)
During the bounty period you should be available to respond to questions, and at the end you want to manually award it. Don't start a bounty and then go on vacation in the land of poor network access. :-)
It's always ok to pop into the site chat room and ask for advice about a specific question/bounty!