We have been to Italy just twice, and covered much of your planned ground. 14 Years ago, late September-early Oct, no crowds. 2014, early to mid May ... Rome and Florence were mobbed. Your itinerary calls for three nights in Rome, but really just a single, full day; you will be jet-lagged the day you arrive. Logistics ... train is good. We did enjoy having a car one week in Tuscany. We picked up a car in Pisa, went to Lucca, Cinque Terre, then down to Sienna. Drove back to Pisa to return the car and then flew out the next morning from Pisa via Milan to Boston with an arrival in time for dinner. Someone else suggested you concentrate your time in Rome; that's a good call, at one end or the other. I suggest that you spend three nights in Rome up front and don't go back.
Open yourselves to the unexpected. Smaller cities have charms and romance that you may choose to enjoy. Orvieto's magnificent cathedral had some wonderful frescoes by Lucca Signorelli and Padova has Giotto's wondrous Scroveigni Chapel. And plan as you might, be open to the unexpected. We stumbled onto a gelato festival when we were in Orvieto. Annother find: at first I was unhappy to have booked three nights in Milan at the end of our 2014 tour, but, on that May trip, we stumbled onto "Piano City" ... a wonderful weekend festival of classical and jazz solo and duet performances in the Public Gardens and other venues throughout the city. Far more enjoyable, if less historic, than Leonardo's "Last Supper." We spent the better part of two days just hanging out in the park and enjoying music.
Cinque Terre will be wonderful; we spent two nights there and I still consider our one full day there as the day I went to heaven.