I am thinking we should rent a car and try to stay in either Montepulciano or Pienza. Would it be best though to stay somewhere closer to Fiumicino Airport the last night? Maybe Orte? Or some other nice town close to Rome?
I would stay closer to the airport the day before the flight. I would recommend a night or two on lake Bracciano, fairly close to Rome airport (45 min drive).
The famous towns on the lake are Bracciano, Anguillara and Trevignano. Bracciano is famous for the castle (and also for Tom Cruise's wedding there). You should probably stay in Bracciano, since it's closer to the airport, but if you arrive early the day before, you may want to swing by the other two towns as well.
I have driven the down the coast. Nice drive. We made it from Lucca to the airport, but it was a long drive. We have also stayed near Ostia Antica. That's fun.
Montepulciano is not terribly far from the Autostrade, but the smaller roads that get you there are by no means fast roads. www.viamichelin.com estimates that you can drive from Montepulciano to Fiumicino in 2.5 hours, but I'd feel more comfortable to give myself double the time. Remember that you need to re-fuel the car before dropping it off, unless you agree to pay the rental company's refueling price. Check-in time for your flight will vary depending on whether it's direct to another continent or whether you make a connection within Europe.