I can only address a few items on your plan. Getting to Pompeii from Rome, you'll take a train from Rome Termini station, Termini to Naples Centrali Station, then Naples to Pompeii "Scavi" stop via the local circumvensia train. You'll need to reverse that train route to get anywhere in Umbria -- though you may be able to train from Naples to an Umbrian destination, but likely routed through Rome anyway -- depends on where you are going. Umbria and Tuscany are 2 different regions and a lot of ground in and between!
Pompeii. If you stay the night in Pompeii that would be most ideal -- from a suggestion on one of these boards, I made sure to get to Pompeii ticket office (June of 2013) 15-minutes before they opened and my daughter and I were the first one's in line and had the ruins pretty much to ourselves until 11:00am, then all the tour busses arrived and it was very crowded, turned a corner and literally hundreds of folks with group leaders holding up their paddle number so those on the tour wouldn't get lost from their group. We got some amazing photos without another person in them before that and it made it so much nicer. For our trip we rented a car from FCO and drove to Ravello on the Amalfi Coast, then got up early one day and drove to Pompeii so it was (relatively) easy. We combined that with climbing Vesevius in the afternoon -- both were amazing and we were fortunate to have a beautiful, clear, sunny (but very hot) day. REGRETS: not visiting the Archaeological Museum in Naples, which is where most all the items from Pompeii went. NOTE: Pompeii is amazing, but don't expect to see frescos and artifacts in place, though there are a few in poor shape and there are some artifacts, but vastly just ruins, but amazing to be walking amongst it and early enough to miss the crowds and hear your thoughts.
On the Car -- we are returning in October of this year, too. 2-nights Florence, 2-nights Venice, and 4-nights tuscany area. We are flying into FCO because the flight was incredibly cheap from Toronto. I thought to train Venice/Florence and just rent car in Florence for touring Tuscany, then train back to FCO -- low and behold renting a car for 3-days in Florence cost more than renting a car at FCO for the entire trip. So, we're renting from FCO and heading straight out. Additionally, and unfortunately, picking up a car in one location and dropping in another will cost you a small fortune. But cost-sharing will divide things up.
I've been using Booking.com to find and secure rooms both last year to Italy and this year and it's worked out very well.
There's lots of useful information on this and other boards so keep searching -- I wish I had saved the links to give you, a few were quite good.