1) Yes, but you will most likely be charged for a drop off fee.
2) Suggest you check AutoEurope.com for rentals. It will indicate various pick up and drop off points for different rental agencies. Many contributors to the Forum use them for auto rentals because of good prices and great customer service.
3) I found it easy to drop the car off near the Tiburtina rail station. It requires minimal driving on local Rome streets. I had the misfortune of dropping one rental near Termini station and would never do it again.
4) You might have to rent your car at an airport location to get an automatic transmission.
I’d also suggest renting with $0 deductible and make sure to get an International Driver Permit at AAA. Familiarize yourself with ZTL’s in Italy to avoid massive fines. Read the “Driving” info in the Travel Tips section.
Happy motoring!