For the record, I appplied to 10 programs, got interviewed at 9 (withdrew from 1), and was subsequently admitted to 7 and waitlisted at 1. I won't say where I applied, but there were mostly all top 10 programs. I originally thought I would get in nowhere, so this was a nice surprise. Needless to say, I feel somewhat well qualified to give application advice.
1) Admissions, contrary to what many say, is NOT random. I did a ridiculous amount of interviews and saw many of the same candidates at multiple schools. Were they just overly lucky? I think not. If you're application is strong, you will be given the proper consideration. The hard part is gaging where in the pack you lie (I didn't know this until well into March).
2) I felt like I applied to too many programs. It's a good situation to be in, yes, but turning down people is very, very difficult. I think 5-6 is a good number. Like I said before, admissions is NOT like the lottery.
3) Focus on what you have to offer in terms of skills in your statement of purpose. Why should the grad school choose you? I was applying right out of undergrad and as sophisticated as I thought my research interests were then, I look back and realize how naive this portion of my statement must have sounded. I spent maybe three sentences on what I'd like to work on. The rest was focused on how my past helped shape those interests and provided me with the skills to pursue research in those interests.
4) I'm under the impression that schools already have a ranked list pre interviews, and that the order is unlikely to change except in extreme cases. So don't fret, and just be yourself.
5) It's not necessarily to contact professors beforehand. It's sometimes useful to do so however, to see if a professor's interest in a particular area not necessarily listed online. Also useful to see if prof is taking new students. Completely optional though, just make it clear who you want to work with in your ststament.
6) write down your thoughts after each day during interview weekend for each school. Make notes of first impressions of professors. You will be grateful come decision time