Acceptance Numbers: It varies greatly. The AAPA website has some invaluable information on each school (www.pathassist.org/?page=AboutUs_NAACLS), plus links to the program websites. 2-..37 or 40 something at Rosalind Franklin now? But I think the average class size is 10-12.
As far as applying to a number of programs, obviously the more you apply to the more chance you have of getting into one. However, I recommend that you look into each program as well as the location to find a good fit for you. You're going to be living there for 2 years! Also consider what the program can do for you when you're job hunting: have you gotten quality rotations? is there an alumni network in the region where you want to ultimately end up?
Research - don't think it really matters.
Education vs Experience - I think each program has a different prioritization as far as grades/experience.
Experience can be a plus if you have worked in a medical field before and understand hospital hierarchies and protocols. However some programs value education just as much. I had a 3.9 GPA and had no medical experience coming from undergrad. But I am educated, intelligent, and trainable. Some programs look at it this way: a high GPA -- especially with As in courses like anatomy, physiology, advanced biology/chemistry -- means you will probably not fail out after didactics your first year and make the program look bad. You'll probably also be great at studying for boards and keep their certification pass rate at 100%.
I've found that my class has a mixture of people with experience vs people with good grades, but it is kind of skewed towards the academics. I wouldn't be surprised if the majority of us had GPAs over 3.5
Shadowing: I would highly recommend it. They want to know that you know what you're getting into and are going to follow through. However, they understand if there are no PAs in your area for you to shadow.