Venice (3n), Florence (3n), Tuscany...somewhere in the countryside (2 n), Amalfi Coast/Positano (3 n), Rome (4 n)
Tuscany - I'm unclear what you are looking for because you say "countryside". 2 nights only leaves you one day to see the town you're staying in. Orvieto is a town (and I highly recommend it) with views of the surrounding hills. I don't remember any views in Siena except from the tower. By the way, Siena is an easy day-trip by bus from Florence. I haven't been to Montepulciano. It sounds lovely (what doesn't in Italy?) but it seems to be a hassle to get there. Consider that you want to go from your Tuscan/Umbrian town to the Amalfi Coast, a long journey with transfers. Just for that reason, I would choose Orvieto because it has good train connections to both Florence and Rome.
I haven't hiked on the Amalfi Coast, but I've read that much of the Path of the Gods is for serious hiking, not leisurely walking. There are no trains to the Amalfi Coast towns. If you want to see Pompeii, Herculaneum and/or Capri, Sorrento is the best base. It has direct ferries to Capri, a local train to the ancient excavations and Naples (the Archaeology Museum is a wow), and buses/ferries to the A/C. If you want to spend your time exploring the A/C towns and enjoying the beautiful coastline, I suggest Salerno as a base. It is on the main Trenitalia train line, making it much easier to get to than Sorrento, and has bus and ferry service to the A/C towns. You can also visit Paestum by bus or train.
If you can extend your trip, I'd add one night to the A/C and one to Rome.