How much time to allow for Vatican and St.Peter's

That is a rushed visit.
The Sistine Chapel is inside the Vatican museums. There are some color coded itineraries which are marked on the ground. You can choose the shortest and it will cover only the main art works (all itineraries will go to the Sistine Chapel). Still you need at least one hour to say you've been inside.
The walk from the museum to the basilica is not short. Half hour will be gone just for the walk. Then you need to hope there are no crowds queuing up to get inside. You need to go through metal detectors to get inside the basilica, so it's like going to the airport.
Once inside there are many things to see and maybe you might want to get on top of the church as well. Also the Vatican Treasures museum is worth it.
You should plan for a late lunch (around 1:30pm, when Romans go to lunch), and there are also good restaurants around that area that are less pricey than around Piazza di Spagna. I'm saying this also because from the Basilica to Piazza di Spagna is a long walk, even if you take the metro, because the Ottaviano station is not near (of course you could also opt for a taxi ride).
Of course some people here might consider it still very rushed, but at least I gave you at least 90 min. more to visit things.

Roberto is right, you need to factor in lines at the basilica as well as the walk through what looks like miles of corridors inside the Museums and then on the street. Near the Vatican is Isola della Pizza, great stop for lunch.

It may be more efficient to reverse the order. Visit St. Peter's first, which you can do even well before 9 a.m. The security lines will probably be shorter that early. Then walk over to the Vatican Museums entrance (5-10 minutes). With pre-booked tickets, you won't have to wait in the long security line. Just show your reservation and go through the much shorter (or non-existent) security line. The only advantage I can see of going first to the Sistine Chapel is that it probably won't be crowded, but since you're looking up, and the guards keep the noise down, crowded or empty doesn't make much of a difference.

Thanks for the advice, I thought my initial plan might be a little too rushed, but now I've got that confirmed. I'll plan for around 4-5 hours to see both the Vatican and St.Peter's.

Chani--I had never thought of the option of seeing St.Peter's first since everything I've read so it's most convenient to see the Vatican first, then end at St.Peter's for the fact that by ending in St.Peter's you have are in a good spot to exit the area. But now that you bring up that option, that could seem like a good idea as we could get started right away in the morning (St.Peter's opens at 7am), so please let me know if this plan sounds better:
1--Eat breakfast somewhere near our hotel (Ripetta 25)
2- Be at St.Peter's at (or very near) 7am when it opens
3- Be done with St.Peter's by 8:30am, and over to the Vatican around 9am (or slightly after)
4- Do the Vatican Museum little after 9am, and be done around noon
5- Walk to Mercato Trionfale Market for lunch (open until 2pm)
6- Take a cab or some other form of transportation back to our hotel to unwind for a bit

Does that sound like a better option given that the fact we can get started first thing in the morning, rather than waiting for the Vatican to open at 9am?

My only other question with that option: given the fact we will be pre-purchasing our Vatican tickets online, how specific are they in terms of the reservation time for your ticket? My only worry with this option is for for example we have a 9:30am ticket for the Vatican, and happen to run long at St.Peter's at don't get there until 10am, what happens then? Do we lose our reservation?

Don't rush through the Vatican - enjoy and appreciate it! After you're finished then figure out what to do with the remainder of the day.