It's a rule in English that "and I" is only used when it's the subject not the object of a sentence. (Subject is the thing doing the action, and the object is the thing being acted upon)
The proper way to say this would be "she's coming over me and my girlfriend's movie night" because "she" is the subject in this sentence since she is the one "coming over".
when you say something belongs to you, you don't call it "I's thing" - you call it "my thing".
if it belongs to your friends Jim and Bob, it's "Jim and Bob's thing" or "Jim's and Bob's thing". likewise, when it belongs to your gf and yourself, it makes more sense to say "my gf and my" or "my gf's and my", rather than inventing a brand new word that sounds so very stupid.