To answer your question, I would not use use "ain't" in written correspondence. Many people use it in the spoken language, but it's not written as much. Even my husband, who will sometimes say it, would probably never write it. It is not considered vulgar, just very colloquial. While it *could* be construed as a lack of education, that's not necessarily the case, since many people from the south use it, educated or not. Some northerners do, too.
"I'm late, aren't I?" is correct, since we don't have a negative contraction for "am not". You could also say, "I'm late, am I not?"