The difference between may and might is very small:
Use may when the event is slightly more likely to happen:
“Shopping! I’m going to buy some new clothes, and I may get a new hat as well.” (it’s slightly more probable that I will buy the hat)
“I might go to the movies. I’m not sure.”
(it’s slightly less probable that I will go to the movies)
However, in this simple case, the words really are interchangeable; you can use either one.
When making guesses about something that happened in the past, we usually use might + have + past participle.
“I don’t know. She might have gotten a promotion – I’d heard a rumor that the boss was thinking of making her a manager.”
When asking permission, use may (or can/could):
This question is correct, but it sounds rather formal. Most English speakers would probably say “Can I open the window?” or “Could I open the window?”
May not means “no permission”; might not means “maybe not.”