It's difficult to give grammatical rules because these are not "real" words and you shouldn't treat them like real words. They are just examples of how people can mumble and slur words together. These are impossible: "Gotta she go now?" / "Hasta she go now?" This is why you shouldn't think of them as words. "She's not gotta go now" is OK but a little awkward. If you want to express how someone mumbles and slurs their speech, then just listen to examples from a real speaker. You might be surprised at how closely they follow regular grammar.

With "gotta" - then all are OK in informal speech except "Gotta she go now?" (don't use) Likewise with "have to", all OK except "Hafta she go now?" We never write "hafta". Writing "gotta" in negatives and questions looks unusual and odd to me but considering "gotta" is such an informal word, I can't see any real problem with doing so, in the right context.

I agree with Peachey. Native speakers don't say "hafta," or "hasta," or "gotta." We say "have to," "has to," and "got to" really, really quickly and don't enunciate our words so they sound like that. The only time I would ever write "hafta" is if I were directly quoting someone and I was trying to emphasize that the speaker was mumbling or slurring their words. If I didn't want to point out something very specific in the a person's manner of speaking, I would still write "have to." You should recognize that people sound like that at times and understand what they mean, but don't use those "words."