MIT:
Can I visit your department or contact faculty before being admitted?
No. Official department visits, including faculty meetings, are arranged after students have been admitted.
Stanford:
Can I tour the department or meet with faculty prior to admission?
No, due to the volume of applicants to our program, we do not schedule tours, interviews or meetings with prospective applicants.
NYU
Does it help to be in touch with an NYU faculty member in advance?
Yes and no (mainly no). Very often a faculty member gets email from an applicant saying that [s]he wants to work with that faculty member. But this isn't possible until the applicant enters the program, takes the core courses, passes the qualifying examinations, and only then is a thorough assessment carried out (including a lot of self-assessment) and an advisor chosen. So don't use this as a way of trying get a professor to support your application.
Of course, it may be genuinely the case that your own interests lead you to interact with one of our faculty members, quite apart from the question of admissions. In that case your queries will almost always be met with a suitable response. The point is that in most cases, it is easy enough to separate a real research inquiry from a message designed to try and obtain admission.
Contacting the staff person in the graduate office is fine.