To get into Rome from the airport, I would just take public transportation, or a taxi to my hotel. If you want to drive from Rome to Florence, pick up your car when you're about to leave. The same goes for Florence. You don't need a car there, and it's in fact a hindrance, so why pay for parking? Just pick up a car when you leave.
I've only ever had two problems with renting a car in Italy. One was that I changed my plans and decided to drop off early at an agency in Rome on August 15th. Huuuuge mistake...even the car rental agencies were closed for Ferragosto. I had to drive the car out to the airport to turn it back in and then had to take a taxi back.
The other problem I had...well, that my husband had...is that he put diesel fuel in a non diesel car we had rented...need I say more???:grin: We had to spend three hours or so in a garage waiting for it all to be drained out. He had a tendency in the beginning of our trips to Italy of wanting to take charge of all the "manly" things like renting the car, and getting instructions on it, and pumping the gas. I'm a woman who picks her battles, so I let him have at it for the sake of his "masculine pride". He used to want to handle the requests for lodgings too, in his "native" American English. Another big mistake in some smaller areas and hotels. By the second trip he let me handle it or at least participate!:grin: