You just made the mistake of mixing formal and informal in the same sentence: "Você" (formal) and "contigo" (informal). It should either be "Tu queres que fiquei aqui contigo" or "Você quer que eu fique aqui com você".
Yes, it would be the same thing. No, one is not more formal than the other. These are just two different ways of saying the same thing and people use them according to where they're from.
"fique" confuses me. Should the present subjunctive not derive from the root of the 1st person singular indicative present and just replace the o/a with e or the o/e with a?
Verbs like "want" or "would like" are followed by the infinitive form, not by noun clauses beginning with the conjunction "that" (different from Portuguese).