You're going to be busy. I hope your group handles jet lag well.
I personally don't think there is a best area of Rome to stay. Certain neighborhoods offer certain things, which may or may not be what you are looking for. You could stay near the Pantheon area, which would be pretty centrally located for getting everywhere you want to go. Other central locations that are popular are Piazza Navona and Campo dei Fiori.
If you prefer a quieter neighborhood, I'd stay closer to the Vatican. It's a little further away from everything but if you use the metro, it doesn't take too long to get from place to place.
Pairing things up in a sensible order might help you decide where to stay. For instance, Vatican/St. Peters should be done together, and not on a day you do anything else time-consuming. Spanish Steps, Trevi, and Pantheon can all be done in a row.