Do you have 9 or 10 nights on the ground in Italy?
This is a critical detail as even 1 night, plus or minus, can affect the itinerary. But even with 10 nights on the ground, you can't effectively do 5 destinations. You'd spend more time packing up, dealing with transport, finding the new hotel, unpacking, getting acclimated to a new place, etc. than sightseeing.
So why have you chosen these destinations, and what do you want to do when you get there? Your interests will determine which of them should stay, which should be given more time than another, and which should be removed from the list. Time of year can matter as well. For instance, the CT isn't a great choice for winter.
Yes, ditch the idea of a car: public transit is the way to go. Italian trains are great; some locations, such as Montepulciano, require a combo of rail and bus.