Should it be awarded in a given time-scale?
If you use an actual bounty, you either award it within a given time-scale (it expires after 7 days, but you have a short grace period during which you can still award it) or you lose it. If no answer meets the criteria, you can choose not to award. However, in that case it might be awarded automatically at half value. These and other details are described in the Mother Meta thread How does the bounty system work?
A PPCG-specific alternative, if you want an open-ended time scale, is to post an answer to List of bounties with no deadline.
Would it be seen as self-promotion? Is it not the 'done thing'?
No. It promotes the question rather than the OP. There is a cost in rep points to bountying your own question, but I've never seen a cost in real reputation (i.e. social respect). If the same person did it every week then maybe that would change.
What's a good amount to put up as a bounty?
That depends on how much you can afford and how hard the criterion is to achieve. I've never seen anyone criticised for stinginess because they posted a bounty at the minimum possible level (50 rep). If the intention is more to have a fun prize rather than to incentivise people to tackle a really hard problem, and bearing in mind that you currently have about 2700 rep, I would suggest picking a value from the lower end of the spectrum.