I have no preference between "Will you.." and "Do you want to be", but I have to disagree with the "girlfriend" / "girl" part. As an American if someone from my side of the Atlantic asked me to "be their girl" I'd wonder if they planned to follow up with "So are we going steady now?". It sounds very old-fashioned here! Straight out of a 50s sitcom.
Fidanzata, compagna, morosa, (l')"amica", amica del cuore, ragazza, donna, musa, dama, principessa, regina, partner, dolce metà, ... - - Fidanzato, compagno, moroso, (l')"amico", amico del cuore, ragazzo, (adone), uomo, cavaliere, principe (azzurro), partner, (-), drudo, boy, toy-boy, ...
I can't reply to your later post, for some reason, so I'll just respond here.
Thank you for the clarification. I've since gotten to the possessives part of the Italian tree and I'm starting to grasp the point. Your comment definitely gave me a headstart in getting it right in my head, though, so thank you. :)
I'm going to be practicing this for more than a couple of days before I've finally manage to cement it, no doubt, but still, it's a good start!
As for the 'cara', I've always understood it to mean 'dear' when used as an adjective (like in 'cara amata'.)
I'm a beginner, I'll take everything under advisement. But okay, I think I can remember that..
Articles are used if it is a thing or a person not related to you (il mio libro), plural (i miei figli) even if they are, or if it's modified with an adjective (la sua cara mamma). So that should mean 'la mio cara mamma'.. is.. also correct?
I'm still a beginner, but what I've learned is that in Italian many sentences can be changed from statements to questions without changing the words. It would be like an English speaker saying "You have flowers," as opposed to "You have flowers?". But in English we might take that as a question of surprise, whereas they do not. They just do it all the time. The "do" that they keep adding to the translations isn't actually in the Italian sentence, but it is the best translation we have in English (another example of a non-literal translation).
Be careful! "Voi" is 2nd person plural ("You all" as you say), while "vuoi" is the 2nd person singular of the (of course irregular :-) ) verb "volere" (to want), so io voglio tu vuoi egli vuole noi vogliamo voi volete essi vogliono
So if it was actual courtesy voi it would have been "Volete essere la mia ragazza?" or, if you were talking to a group of girls "Volete essere le mie ragazze?"
So the translation is correct.
You are correct saying that courtesy voi is no longer used in everyday life but you may find it watching historical movies refferred to kings/queens/bishops or just among noble people...just for you to know.