Are university lawyers usually faculty members or “just” 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.