What are the circumstances which determine whether a question Asker can close or delete his question? If the Asker hasn't got a good answer even after setting a bounty, should he delete his question or just close it? What happens to the reps in this case?
That's all pretty hypothetical isn't it?
Normally you'd assume you only ask a question because you want an answer and have a vested interest in doing your best to get an answer. So that means no, you should not be deleting it because you'd would want to wait to get an answer even if it takes a long time.
Besides, even if you learn to live with it or find an alternative solution, the eventual answer might still benefit someone else.
If your question is very localized, for example because you corrupted your OS and that was causing the issues and reinstalling the OS fixed it. In that case getting the question closed would be appropriate.
So really what you should ask yourself: "Does this question provide a lasting value to somebody on the internet?" If the answer is yes, then you should not delete it.
So posts with varying degrees of poor quality would not pass that test and therefore could or even should be deleted to keep things clean
Neither. A question's asker cannot delete a question after answers have been posted or a bounty has been offered. In the case that a question was asked and is fundamentally flawed (but isn't obvious from the perspective of the community) but has received answers, you can flag the question for moderator attention and explain why it should be deleted.
In any case, though, unless the question is just plain nonsense, consider leaving it and allowing the community to maintain it; if it's not suitable, it will be closed and likely ultimately deleted. Otherwise, it could still help someone else solve a problem.
If the issue is simply that the question hasn't gotten any answers, then leave it. Like above, if it's too narrowly scoped, it'll probably end up closed. If not, it will fall into the dungeon of unanswered questions and get occasionally popped to the top, or someone may search and find it.