I found this site to be helpful regarding clitic pronouns: https://duolinguisto.wordpress.com/category/clitic-pronouns/
I put "he is between you and me" and got it right. But in other forums people have noted "stare" is "to stay". In the DL tinycards set, it's also "to stay." But when I run your sentence through translators, it translates to "sta" for "stand". So, I can't say why your translation would be wrong. But as far as your question about a meaning using "stay," I thought it made sense in terms of a relationship. Like if a man and woman were dating, and she still had a thing for her ex-boyfriend, I could see the current boyfriend saying that "he stays between us"--as in, his memory seems to still be with you in a way that is obstructing our relationship. But I don't know, as I just used "is".
I am learning Italian and am a native English speaker. In English, at least, "you and I" would be incorrect for the same reason you wouldn't say "He stands with I." "Between" is a preposition that requires two--but plug in another preposition (as I did with "with"), and you'll see that "I" isn't correct, because the self-reference is objective, not subjective. Something is done "to me." But "I" do things to others. "I stand with you." But "You stand with me (not "with I").