Itinerary help: Florence, Tuscany, CT

You would do better to visit Florence first and then stop at Lucca on the way over to La Spezia and the Cinque Terre.
When you leave the C/T, take the train back to Florence. Pickup your rental car there to the short drive down to the Tuscan hilltowns.
Many travelers turn their rental car in at Hertz in Orvieto and take the 70 minute train ride into Rome Termini. Flights out of Italy are best taken nonstop out of Rome FCO. Florence Airport is very small and not preferred.

We did the trip from the Cinque Terre to Siena by train several years ago and it was quite the ordeal: 4 trains, almost 4 hours, and we had to stand up on two of them due to crowding. So renting the car in La Spezia is a thought. I would do that next time. You will need to stay somewhere with parking for Siena, of course, if you forego the train. Not an impossible task but something you need to bear in mind.

I can recommend a charming agriturismo just outside of Montepulciano, Poggio-Etrusco.

Your home airport is MCI (Kansas City). MCI, in April, has connections only to the following international hubs that have non stop flights to Italy:
ORD (American to Rome FCO)
ATL (Delta to Rome, Venice, Milan)
However the American flight ORD - FCO is seasonal, and so are the Delta flights (except for the one to Rome, which operates always).
So, based on the above, the best flight option for you, which entails only one stop, is with Delta to Rome, via Atlanta.
Since there is a high likelihood that you will be flying in and out of Rome with such flight, I would do:
Arrive Rome, and immediately head to Florence by train (2hrs trip, tons of trains from morning to evening)
Visit Florence (3 nights)
Take train to Cinque Terre
Visit Cinque Terre (3 nights)
Rent a car at La Spezia, drive to Montepulciano/Pienza area.
You can visit Lucca along the way. Park outside the walls.
Visit Siena and Val D'Orcia area while in Montepulciano. No need to spend a night in Siena, since Montepulciano or Pienza are close. Stay at least 3 nights in that area of Tuscany.
Drive down to Rome. Return the car in Rome as soon as you arrive. Spend the last 3 nights in Rome.
Taxi to airport on the morn of departure and return home via ATL (Delta).

I used 12 nights in my itinerary. I would Allocate the 13th to either Tuscany (Montepulciano/Pienza) or Rome. But It's up to you and your personal preferences.

If, as you say, you enjoy art and architecture, then I think you are short-changing Florence. It looks like you are allotting only one full day and two part days to that veritable treasure trove.