In does he not, the verb is does, the subject is he, and the rest of the sentence begins with the word not in its regular position. The same with doesn't he: now the verb is doesn't, so it comes before the subject.

Is he not grammatically correct?

Whether "he don't" is acceptable in a particular dialect is a dialect specific issue and should be addressed as a separate question. Whether "he don't" is acceptable in General American English, the answer is no, it is not.

Does not or doesn't grammar?

DON'T (do not) is the negative form of DO while DOESN'T (does not) is the negative form of DOES. The tiny difference is: - Use DOES / DOESN'T if the subject is third-person singular (he, she, it). - Use DO / DON'T everywhere else.