Oh Wow The Palio! is in Siena then. Well that would be an amazing experience but also a mob scene you probably don’t want to deal with. If you can get to Siena for a day trip before 7/2 that might work but it's still going to be rather hectic there.
OK- some comments, some suggestions on your new plan.
Hate see you only have 3 nights in Rome but will leave that for now- it does solve the car rental drop off problem
Just rent a car once, not twice. I’ll show you how below.
Florence- I really encourage you to stay in a hotel for your first 3 nights. As first timers to Italy/first stay you will find having a 24/7 desk, bkft included, etc invaluable. Renting an apartment for just 3 days and on your jet lagged arrival day is not usually the most efficient thing to do.
It is going to be hotter than Dutch love, so I think you might want to get out of the city/away from crowds after 3 nights in Florence
Orvieto- maybe on your way to Abruzzo, otherwise too far for a day trip. Civita not necessary.
OK- here is my suggested itinerary, just a few changes to yours:
06/26 -Day#1 -Friday- arrive FCO-train to Florence
sleep Florence - 3 nights, look for hotel near center/train station. Keep it simple. Florence is compact, you can walk everywhere
06/27-Day#2 - Saturday -Florence sight seeing - Sleep Florence
06/28-Day#3 - Sunday - Church - Renaissance Walk/Picnic - Sleep Florence
***If you do want to visit Lucca/Pisa then I would stay 4 nights in Florence, 2 at agriturismo. Take train to Lucca/Pisa.
06/29-Day#4 - Monday - Agriturismo stay 3 nights (or night #4 in Florence- day trip to Lucca here, or Siena if you are brave!)
Make sure the place you pick is not far from highway, easy access, parking.
Pick up car as you leave Florence. (We picked up at Hertz Sansovino location- outside ZTL, easy drive down Chianti, there is another Hertz just as easy that Roberto mentions. Look for those posts)
Choose a hill town maybe 2 hours south of Florence- we loved Montepulciano, Pienza is also very nice. Find an agriturismo in that area- a pool would be welcome. You can day trip to multiple towns, Montalcino, San Quirico, Pienza, Montepulciano, Bagno Vignoni was fun, etc.
Tuscany is quite easy to drive around, get used to driving on these 3 days- you will be driving from here to Teramo.
06/30-Day#5 - Tuscany
07/01-Day#6 - Tuscany
07/02-Day#7 - Thursday - Drive to your lodgings in Teramo - should be pretty straightforward- 4 hours?
(get a good Michelin map- amazon.com is your friend)
07/03-Day#8 - Friday - investigate Abruzzo Region and ancestry
07/04-Day#9 - Saturday - More of Abruzzo
07/05 - Day#10 - Sunday - Church/beach/picnic
07/06- Day#11 - Monday - Drop off Car & take Morning train to Rome - 3 nights- stay central- will save you time/touring.
( if you feel comfortable just driving to Rome to drop off car that would save a good bit of time getting you to Rome much earlier in day. You've only got 2 full days there now!)
07/07 - Day#12 - Tuesday - Rome
07/08 -Day#13 - Wednesday - Rome
07/09 - Day#14 - Thursday - Flight home @ 11:50hrs - plan getting to airport @ 0800hrs. Taxi from center is fixed 48€