Can most functions of universities be categorized as teaching, research and service? I am writing a research proposal and I plan to approach it by dividing the most important functions of university into three categories--teaching, research and service.

Teaching will include all forms of education afforded to students.
Research broadly refers to all attempts at generating new knowledge.
Service refer to service to the monarchy, state, the public and in some situations, to the students as customers.

Do these categories sufficiently cover most functions of a university? Is there anything important that I could not include in these categories?
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