Are university lawyers usually faculty members or “just” support staff? Many universities will have a Faculty of Law whose academic staff would be largely (or entirely) composed of certified lawyers; I imagine that the same would be true of many of their PhD students.

Similarly, many universities will have dedicated legal teams that assist the university staff with dealing with legal issues (e.g. corporate  compliance issues, patent law, etc).

Would the lawyers in these legal teams typically be composed of staff members and/or postgraduate students from the Faculty of Law, or would they simply be lawyers that have been hired by the university as support staff?
Large or well-funded universities have dedicated general counsel like any other corporate entity, and they function in the same way that corporate general counsel do. Their job is to protect the university's legal interests, which includes things like negotiating with unions, damage control and prevention for legal problems, and defense during litigation, which is often contracted out to a specialist law firm depending on the nature of the litigation. In particular, they will often be on the other side of the table from faculty, so it would be problematic if they were faculty as well.

I suppose technically the office of general counsel is "support staff," but it's a bizarre way to refer to an office that works at the very top of the administrative hierarchy.