1. Don't think so. I applied to 5 schools, no interviews.
2. I had a faculty mentor assigned to me on arrival. I ended up finding someone more related to my area of interest later anyway.
3. Of course. MPH applicants come from a variety of fields. Some programs have basic math requirements but even that seems rare.
4. It didn't hurt me. Unless that's why I didn't get into UDub... Seriously though, it's a professional program that seeks out diversity in its student body. There are kids fresh out of college, foreign doctors, mid-career changers. I think you're judged based on what you can bring to the class. That being said, rock those essays and find profs who really support your application.
Where you apply is going to depend on a number of things. You'll need to pick a concentration and look at schools from there. At least in my current program, it's not easy to switch concentrations once you're here. After I decided on my concentration, I made a spreadsheet of schools and had their admissions statistics. I ranked them according to a variety of categories (strength of program, cost, location, etc) and picked 5 to apply to.