I don't know the answers to all of these, but here's a shot at a few.
- Is there an auto-reject mechanism? (e.g. throw out low GPAs or low test scores or people without certain core coursework before even looking at the recommendations and the statement of purpose)
I was told by senior faculty at a top 5 school that this was indeed the case.
- What is more important for admissions: Statement of Purpose (i.e. your future research interests) or Writing Sample (i.e. the quality and thematic choice of your current work)?
I would say that the
SOP is more important. Other factors in your apps (recs, gpa, gre) should already give them a good idea for your research/analytical/writing abilities. You need the
SOP to convey what specifically you are interested in researching, why you are prepared for such research, and why school X is the best fit.
- Who will be the first to have a look at your application: graduate director/chair of the admissions committee or a randomly selected faculty member? Or a person in an administrative function?
I'm sure this depends on school, though the programs I know of either had "a person in an administrative function" or not so randomly selected faculty members (based on what you put down in your app as being your subfield.