This answer is based on my experience in the US. It won't necessarily apply to every institution.
Typically, these titles are used for people visiting University X for an extended period (one or more academic terms) to work on a research project with a faculty member (call her Prof. A) who is already there. If this is not your situation, then please clarify your question.
You would not be employed by University X, but would typically be entitled to courtesies such as library privileges, computer access, and perhaps office space. The exact qualifications needed for each title may vary, but if Prof. A wants you to visit, she can probably arrange for you to get whichever one is appropriate for you.
I would expect that if you are not attending classes, you wouldn't pay tuition or fees. There might be a nominal fee for a library card or something like that.
Teaching assistantships would generally be reserved for University X's own enrolled (degree-seeking) students. It is conceivable that you might be able to get one if they are really short staffed, but probably unlikely.
If you want to get paid for teaching, depending on University X's policies and needs, you might be able to get an appointment as an adjunct professor or instructor. This would be something for Prof. A to bring up with her department chair. At some places an MSc would be sufficient for such an appointment, but other universities might require a PhD.
The most likely way to get a research assistantship would be from Prof. A's grant funding. It would depend on the specifics of her grant, the rules of the funding agency, and the rules of University X. In any case, her own official graduate students would probably have priority. Again, this might depend on how strongly she wants you to visit.