Yes, you can find the schedule to Poppi on the Trenitalia site. I tried to pull up a sample here for you:
Don't look to far ahead on the calendar as you won't find options.
Also, since you are unsure as to your flight arrival time into the airport (and also once you get bags,etc) I would just look at the Termini to Poppi flights. As I said, you don't need to book these now anyway. You can wait until you arrive and purchase at the Leonardo station.
As for the car: Yes, it is very easy to pick up at Rome airport (FCO). Try www.autoeurope.com for options. They are a broker for the major brands and tend to offer the best deals. You could also rent in Arezzo. There is a Hertz office directly across the street from the trains station. However, if you are arriving there on a Saturday after 4pm or on a Sunday, they are closed. There is a Hertz office in Arezzo but you would have to take a taxi there from the train station (about 10 mins by car), it is in a rather industrial area.
Yes, if your flight leaves at 11:35am, it is best to spend the night before in Rome. You can either find something right downtown Rome or there are two options out at the airport: Airport Hilton and Hilton Garden Inn. Both have free shuttles to Terminal 5 in the morning. You need to be at FCO at least 3 hours before your departure. The nice thing if you decide to stay at the Airport Hilton is that you can return the car that day/night before, walk from the rental return (it is the large parking garage attached to the Terminals) and then walk the catwalk flyover directly to the Hilton. They have a pool (Garden Inn does not), bar, restaurant (albeit not the best food you will have all trip), gym and best of all, a free shuttle that runs every hour each way to downtown Rome (you can pick up the schedule at the hotel front desk). It drops you very near the Vittorio Emmanual monument (near the colosseum). So you can spend your last night in Rome, eat dinner there, take the Hilton shuttle back to your hotel and be conveniently placed for your next day flight.
Are you going to Casa Ombuto?