I have always used and heard the verb μένω meaning to live. Eg που μένεις - where do you live? I know it means “to stay” as well but I’ve never heard this alternative to the verb to live being used in Greece.
It's not too common of a phrase, that's true. But it's not incorrect. I've heard it being used a few times, mostly in cases where more emphasis is needed (if, for example, someone has stayed with their aunt for a very long time). ^.^
με is "with".
την is "the". Except that in English we do not say "the their aunt", but Greek does.
Note the difference between μ - υ - ν.
You can't; they're pronounced completely identically.
In this sentence, only one of them makes sense because it's the subject of the verb ζουν -- it's αυτοί ζουν but it would be αυτή ζει.