You are discovering one of the challenges of planning a trip to Italy: where to draw the line. Setting priorities is key.
Are Avellino and Siano absolutely imperative? They are challenging to get to. You could do something like this:
Arrive Rome -- assuming it's early in the day, before noon -- and proceed directly to Sorrento via Frecciarossa high-speed train. Use Sorrento as a base for 3 nights. See Pompeii and/or Herculaneum one day, and take the Amalfi Coast bus the next. Sorrento, 3 nights (It is not worth going all this way for less than 3 nights.)
Rent a car and drive to Avellino, a one hour drive - spend 2 nights. Visit Siano 30 minutes away. Avellino, 2 nights. Of course, you could stay in Siano instead and visit Avellino.
Paestum is interesting, but I would save it as it is a bit out of the way and your time is short. Instead:
Drop the car in Sorrento. Take the Frecciarossa high-speed train to Florence. Florence, 3 nights. I would forget Pisa, but if you must you can daytrip there from Florence. EDIT: Drop the car in Naples. Much more efficient than Sorrento.
Take the high-speed train to Roma and finish your trip there. Roma, 4 nights
Much as I love Venice, unless you can sacrifice the Avellino & Siano time, I would skip it this trip. Plus, Venice really needs more than 2 nights. You will be back!
Alternatively you can skip the south entirely and as Ken suggests, work your itinerary around Venice, Florence, & Rome with maybe a stop in Orvieto. (Venice 3 nights, Florence 3, Orvieto 2, Rome 4, in that order.)