If I asked 5 questions and answered them all, and then accepted them, would I have earned the Rep Hunter hat?
No, this would not have worked.
Source: I've tested it a couple of days ago.
Sometimes your own things don't count
In some cases, there may be an unmentioned requirement for certain things to be outside your control, lest it be too easy to get a hat. In particular this is often the case for hats involving accepted answers where self-answers are excluded.