Native here. "Passer du temps avec toi" is still kind of intimate like it's some kind of duty. "Hang out" is a phrasal verb in English which is something that doesn't exist in French. I would say "faisons quelque chose" which means "let's do something" but it implies doing something together. It depends on context and tone of voice.
It's from the verb 'vouloir' to want. Je veux, tu veux, il veut, elle veut, nous voulons, vous voulez, ils/elles veulent.
Since 'sortir' is in the infinitive 'to go out' then the bit that precedes it is more likely to be a conjugated verb than a pronoun such as 'vous'. Also 'veux' and 'vous' don't sound exactly the same. There's a vowel sound differents. Unfortunately veux and veut do sound exactly the same so you'll need to use the pronoun (je/tu/il/elle) to work out which it is.
So 'Je veux sortir' - I want to go out