Can most functions of universities be categorized as teaching, research and service?
Universities, and their students, do a lot of community service. While this might fall under your service category, I wouldn't describe it as service directly tied to the monarchy, state or "public".

At a minimum I think your list is missing


Social: At the undergraduate level many universities actively promote social interactions via greek life which I would not describe as teaching, research, or service. Even at the graduate student level the social networking that students engage in is a critical part of university life.
Sports: Many universities, especially in the US, also stress sports. 
Fund Raising: Unfortunately a large portion, at least in the US, of the university function is to promote the university by raising funds