When someone tells "do this" or "don't do that", it helps to know the context and the reason for the choices. There are pros and cons for most options. It is up to you to decide what is relevant in your case.

Picking up a car at airport give you the most car choices. If you must drive an automatic, FCO is the best place to find such car. However, the cons is that your driving to Montepulciano is more than 2.5hrs vs. if you had picked up your car in Orvieto which is less than 1 hr from Montepulciano. Whether one is ok with driving long distance after land depends on individuals and travel situations. Some only needs to take an 8.5hr direct flight from NY and says it is no big deal, why others might have to fly 23hr to get to FCO and have no desire to do any long distance driving, in which case, taking a train to, say, Orvieto, stay overnight or drive 1hr to Montepulciano might be more attractive even though you are more likely to drive a stick-shift car.

If you need to ask if Florence is a must, I presume you are not into the Renaissance. In which case, you need to answer what you want to get out by visiting Florence. There are many places in Tuscany you can enjoy food and wine . If you decide to include Florence, entering Florence with a car is problematic due to a large area of restricted driving zones, ZTL, for which you will be hit with breathtaking traffic fines. Visit Florence before Montepulciano or after but not during. Before means upon landing in FCO, take train to Florence, and when you are done, rent a car as you exiting the city to head to Montepulciano. Do not drive into Florence unless you really understand how to avoid the ZTL.

With only 10 days I would plan for 5 days each in Rome and Tuscany. The car rental depends on your tolerance for jet lag and your driving confidence---do you sleep on long flights?

I would see Civita as you transition back to Rome and perhaps drop the car in Orvieto---hopefully not on a Sat or Sun.

Montepulciano to Lucca is 2-3+ hours each way depending on the route you take (highway v. secondary roads) and assuming you make no stops. Too far for a day trip, IMO. Even San Gim is about 2 hours each way if you take the 'scenic' route. Some people stick to the highways to cover the most miles in the shortest time. The destination is the priority. We do the opposite because for us the things we do/see along the way are as important as the final destination.

Since you haven't been to Italy before, consider not starting your trip with driving. Different country, unfamiliar car, jet lag... that's a lot to adjust to in the first hours. But if you've had experience driving in foreign countries, then you have an idea of what you're getting into and know your abilities and limits. However, if you must have an automatic transmission rental, greg has pointed out the advantage of starting at FCO.

If it were my trip, I'd spend the first days in Rome, then train to Orvieto or Chiusi and pick up the car. After the time in Tuscany, return the car at FCO. If the outbound flight is late enough in the day, you could spend the last night at Lake Bracciano which is less than an hour from FCO.

We flew into Rome, then took a train from the airport to Rome Termini station and from there a train to Orvieto. We got into Rome at 2 PM and found that do-able. We stayed at Ripa Medici B&B and it was fantastic. Loved Orvieto; definitely worth a stop. Picked up car at Hertz and drove to Civita di Bagnoregio for the morning. In Tuscany we stayed at Frances Lodge outside of Siena. It was centrally located for day trips and only took a few hours to drive from Orvieto. Check my trip report by clicking on my name. We dropped off the car at the Florence airport, which was very confusing and I would NOT recommend. You could take a train back to Rome from Siena I think. We flew out of Genoa so we wouldn't have to backtrack (our last stop was Cinque Terre)