Hey Maksim!
1. You are correct - 'ain't I' is extremely informal (and regional). We would never use it in writing, and most English speakers don't even use it in speaking.
2. 'Am I not' is correct, 'Am not I' is never correct. This is because to turn the statement 'I am not' into a question, the only thing we need to do is flip the subject ('I') and the verb ('am'). Therefore, 'not' doesn't change its position. 'Am I not' isn't arrogant per se, but is often used in a rhetorical sense. For example, if someone says 'I am a teacher, am I not?', what they really mean is, 'Obviously I am a teacher!'
3. 'Amn't I' doesn't exist. As you have noticed, it looks (and sounds) very strange, and English speakers agree! This is why we choose to say 'aren't I' instead, because it is much easier to pronounce and looks nicer written.