We did the Pantheon and the Forum with the
ABD, though not a lot of time in the Forum. It was closing as we got there - Dusty was literally holding the turnstile for us. I think closing is timed with dusk, so if you are not going in December, as we did, you will surely have more time there!
I believe we met at 7:30 for the evening Vatican tour. It might have been 7. Not late enough to eat before, I remember that! We did have the entire afternoon and early evening on our own, so if there are earlier masses, you'd be fine.
We did not climb anything at St. Peters, so yes, do that on your own. If you are doing plenty of other stuff on your predays, it might make the most logistical sense to plan to go back and do that after the Castel Sant'Angelo, when you are let loose on your own. This is the day you do the interior of St. Peter's and also the day you have your night tour. (Day 3, Let's Be Civilized) That way you avoid lots of backtracking, and on your predays you could do more in-city stuff.
We really enjoyed the Capuchin Crypts, which I don't see on your list. Well, maybe not "enjoyed" but we did find them fascinating and worth an hour!
This is a fabulous Adventure - you will love it!