"Mayn't" isn't used very often, but it's a real word. "Mightn't" is used more often, and it's also a real word. :) I wouldn't use "may" in place of "might". "May" is about permission, "can" is about ability, "might" is about probability. The three are often mixed up, which makes the result harder to parse.

Mayn't and mightn't aren't words

"It might rain this evening, mightn't it?" This is perfectly correct. :) "You may go out tonight, mayn't you?" This sounds like a regional dialect from somewhere. I wouldn't accept this as standard. We would normally use a different tag, because "mayn't" is very uncommon (like using "amn't"). I think in this case, you are pushing the tag question rules too far and making sentences that don't sound natural. If you get to a point where a grammar rule doesn't seem to work, then don't use that rule. :)