Ciao Rassell!
Driving from Rome to Siena (and then on to Florence) at the end of the year should not be a problem in terms of safety. You do drive through the Apennines and should make sure your rental car has snow tires or chains in case you were to run into some snow on the drive up, but aside from that part of your drive, you are unlikely to find any snow in Tuscany.
It only snows in the higher elevations along the Apennines (above Arezzo and to the east of the Casentino valley, just to try to place the area), but since you're heading to Siena and then on to Florence, you will not be in areas where it snows. There have been unusual occurrences of snow the last 2 years, but they have been really exceptional and they were a matter of 1-2 days of snow and that was it for the winter.
As for your drive to Siena: are you planning to take it slow and want to make several stops along the way? Traveling with a small child you likely don't want to make the drive too long but I understand a stop halfway might be the best to break the drive. On this drive, I often recommend stopping in Orvieto which is in Umbria, about halfway between Rome and Siena. It is a very charming small hilltop town built on an outcropping of tuff stone. Many of the buildings in town are made with this rock and there are extraordinary views of the town and the surrounding countryside. The cathedral is beautiful and it could be perfect for a break and lunch.
Siena is built on several hills so if you stay in the town you'll likely have good views all around.
We personally stayed at
Locanda di San Martino when we went about 2 years ago and our room was on the top floor from which we had beautiful views over all of Siena. It was also pretty close to the walls so that we were able to park outside the ZTL and then walk to the hotel (avoiding fines since the center is a ZTL, limited traffic area for residents only). You can of course also head directly to a parking lot - the
Porta Romana lot for example is the closest to the Locanda San Martino. Look at the
map of all parking lots in Siena.
Take a look at other
hotels in Siena and
B&Bs.
Siena is a beautiful town - I highly recommend making the climb up the Torre del Mangia, the tower of the city hall building from which you'll be able to see all over Siena and down into the Piazza del Campo. Go to the
Duomo and enjoy the mosaic paintings on the ground - stunning!! The same goes for the Piccolomini Library. We recommend getting the entire Siena Pass so that you can also visit the crypt and the baptistery, truly worth seeing. The Pinanoteca or picture gallery is also very nice, but you can see how it goes with your toddler if he has the patience to endure all this time indoors