help with itinerary - last minute trip!

You have 10 nights in Italy and are trying to fit 4 locations.

Consider heading right to Florence on arrival and putting all Rome nights at end
There is a direct train FCO to Florence at 1:53 that might actually work for you - if not just take Leonardo Express to Termini and grab next fast train to Florence

Arrive FCO- train to Florence
Florence 2 nights
Pick up car as you leave Florence
Siena is a good base 3 nights ( or just outside Siena)
Return car in Siena, take train to Venice 2 or 3 nights
Train to Rome 2 or 3 nights
Fly home

It’s a fast paced trip with bare minimum stays.
A 3 night stay really means just 2.5 days. Changing location takes up at least half a day
Think hard about what kind of trip you want. You might decide to stick with just Rome, Florence and Tuscany
Renting a car for much less than 3 nights is almost not worth the trouble.

Search this forum for info on driving in Italy. All drivers must have an IDP
Read up on ZTLs, parking, speed cameras
Make sure you have a designated driver if you are wine tasting
The alcohol limits in Italy are much lower than US

Trains won’t be a problem
Purchase when you are firm with itinerary
Purchase from official site:
https://www.trenitalia.com/content/tcom/en.html

Ditto all of the excellent advice above!
You can easily hop on a train to Florence on arrival, then work our way back to end in Rome, saving you from the one night stay.
I would also consider three stops versus four. More people generally take longer to do things.

Hi. I agree that you should immediately head to Florence upon arrival in Rome and limit your stops to a maximum of three with only 10 nights. There are so many great day tours from Florence going to different areas of Tuscany that I would choose to make that my base. I would do 4 nights Florence, 3 nights Venice and 3 nights Rome in that order. What a wonderful time you will have with your family!

Is Venice a must do on this trip? If you haven’t booked your flight yet, look into flying into Venice or Florence, then Tuscany and flying out of Rome. This cuts down on backtracking and you may end up with a bit more time to enjoy Tuscany. The cost should be about the same as the round trip flight. Another option, if you decide on the round trip to Rome, take the train to Venice on the day of arrival, then Florence,pick up the car there, do Florence and drive to Rome to finish there, or return car in Florence and take train to Rome. Some towns to stay in Tuscany, Siena, Chianciano Terme, Montepulchino, Volterro.

Generally it is recommended not to buy pre-purchase train tickets on day of arrival because of potential for flight and airport delays (I remember once waiting nearly two hours for checked baggage to arrive at Rome). The fast trains between major cities run frequently. Otherwise do pre-purchase, but realize if you miss a train you will need to purchase new tickets at full cost. Use the train company websites, Italio or Trenitalia, not a third party, and make sure you input the Italian city name, even on the English language website. Finally, strongly advise you get a good paper map of Italy, I use a Michelin map 1 cm equals 10 kms, so you can get some idea of distances, how to get from A to B etc.

I would do Florence 4 nights, Venice 3 and Rome 3 and would skip the Tuscany rental car portion. But from Florence you can do a day trip by bus to Siena which is about one hour away and another day you can leave mid afternoon and go to Pisa to see the tower, cathedral and baptistery. Pisa is about 1.5 hours away by train. You don’t need the full day in Pisa. Pisa and Siena (and Florence) are in Tuscany so you’re getting some Tuscany.

I always use Booking.com and the apartments I get through them always hold your luggage if you get there before check-in time. Not sure about Air&b apartments.

You've gotten some great advice but I'll add a bit! I agree with saving Venice for another trip unless you can fly in Venice and out Rome (or vice versa). The other outlier is the Tuscany driving portion but since you said you want to see countryside...that's all the countryside in this plan! I love the idea of arriving in Rome and training straight to Florence.
When you're done in Florence, get your rental car at the airport and drive to your agriturismo. We're doing a similar plan in the fall and we're staying outside of Siena. So then it's easy to do a day/day and half in Siena, then two days to drive around seeing Montepulciano, Siena, the Crete Senesi or other sights you prefer in the area. Then return your car back to the same airport and take a train from Florence to Rome to finish out your time.
So Florence 3 nights
Tuscany 3 nights
Rome 4 nights

ps: a hotel or b&b usually let you store luggage if you're checking in later that day. Can't say for Airbnb we haven't had great experiences with them in Europe.