What are the trade-offs of working in the office versus elsewhere (as a professor)?

I think it's important to set boundaries with students, with colleagues, and with yourself about when and where you're available for meetings. It's also important to find environments that most effectively support different types of work, and give yourself permission to use them.

Like you, when I'm in my office, I expect to be interrupted; so when I'm working at my desk, I can only productively work on tasks that survive interruption. Put bluntly, the office is where I have meetings; if I need to think, I find a whiteboard in an empty conference room; if I need to write, I go to a coffee shop. (Suresh is correct; I am in a coffee shop right now.) As Daniel says, all three places allow for productive work, but of very different types. Even in the computer science building, for small meetings where I don't want to be interrupted, I prefer to go to the other person's office. And because my undergraduate office hours are occasionally very popular (especially right before exams), I don't hold them in my actual office, but in a larger room down the hall with couches and whiteboards.

You express two points of concern, which I'll exaggerate:

• My students won't like me if I'm not available on their schedule. I agree with DQdlM and Nate. Spread out your office hours to fit as many students' schedules as possible, be in your office (or "office") for every minute of office hours even if nobody shows up, and be willing to offer occasional off-schedule meetings. It might help to announce in your syllabus times that you're willing to schedule sporadic meetings. ("I'm also available for occasional meetings Tuesday or Thursday afternoons; send me email to set up an appointment!") Consider moving (not adding) your office hours if student demand doesn't match your announced schedule. But then stick to your guns. Yes, some students will be unhappy, but that's inevitable; don't take it personally. Your availability outside regular office hours will not be the most significant bit in your student evaluations.

• My colleagues won't like me if they don't see me in my office. I agree with DQdlM and Suresh here. Yes, it's important to be visible and active citizen of your department; that's not the same thing as being constantly on call. The amount of time you spend at your desk will not be the most significant bit in your tenure evaluation. The danger is not that nobody sees you in your office, but that nobody knows what you're doing. Give regular talks to your colleagues and their PhD students showing off the results of your out-of-office effort. Go to faculty meetings, and occasionally offer an opinion. (Careful, that gun is loaded.) Attend seminars, especially for faculty candidates, and ask questions. If there is a regular departmental social event ("Tea" in many math departments), be there. And so on.

Finally, I strongly encourage you to raise these concerns with your department chair or your senior faculty mentors. (You do have a senior faculty mentor, don't you? If not, find one!) They can help you navigate your department culture far better than Some Guy On The Intertubes.