I also think it's best for you to fly round-trip Rome and start your visit in Florence. Do not underestimate how much orientation is required and how exhausting and frustrating it can be, especially since this is your first trip to Europe.
Rome is a very big, bustling, noisy city, its sights are spread out, and there are multiple means of local transport to figure out (tram, bus, metro). I have often had trouble finding my way around Rome - it's often hard to find street signs, streets that run fairly straight change names, many curve, and the city's built on hills. Bus routes go every which way so you have to know exactly which bus to take and where it stops, large metro stations have multiple exits - use the wrong exit and it may take you several minutes of wandering to figure out where you are and how to get to where you wanted to be! Not to mention how difficult you may find it to cross a busy street. Also, Rome's sights are major. You need to have a clear head to take it all in. Yes, it's a real schlep to get come off a long flight and still have to take a train to Florence, but it will be easier than trying to see Rome on your first day. All of the sights you've listed are in easy walking distance of each other, as well as the train station and many other sights. Most of that area is off-limits to most vehicles, so there's little traffic noise and it's pleasant to stroll around. It's a great place to get acclimated, relax and soak up some Italian atmosphere over a glass of wine, a cup of coffee or some gelato.
If you're planning to have a car for all of Tuscany, it may be best to rent the car in Florence and return it there, then take the train to Rome. Driving in Rome is definitely not recommended! And Florence has the best rail connection to the capital.
Do your homework about driving in Italy. Learn the international signs, the rules of the road, and the dreaded ZTL rules. Know that you're likely to pay a premium for an automatic, that the bigger the car, the harder it will be to park, and GPS is a godsend. Many people think Italian drivers are quite aggressive.
Lastly, redo your itinerary like this sample:
Day 1 arrive, take train to Florence (3 nights)
Day 2-3. Florence sightseeing, 2 full days
Day 4. Pick up car, drive to Lucca (6 nights)
Day 5-9. Sightseeing in Tuscany, 5 full days
Day 10. Return car in Florence. Train to Rome (5 nights)
Day 11-13 Sighseeing in Rome, 3 full days
Day 14 Fly home.
This way it's clear to all exactly how much time you're spending in each place. Note that you "lose" about 1/2 day every time you change locations.