Margaret:
Staying in an apartment in the Rome area is a good option. We stayed at a B&B 20 miles south, and got to visit a number of great tourist sights in the Rome suburbs; www.olivetreehill.com We took the commuter train into Rome.
I agree with the others in that taking a train to Florence would be much more efficient. A car is not needed within the city. If you want to take a day trip to the hill towns 40-50 miles south, you can easily rent a car in a city location and return it later. Your wine tour has already been addressed. Florence and Tuscany are worthy of more time than you have allotted there.
It's quite a distance over to Nice or Monaco, and quite out of the way. If you want to see a breathtaking scenery, just take a train over to the Cinque Terre cities, west to Pisa and up to La Spezia. You would take a regional train into Cinque Terre, which is not a place for automobiles.
From Cinque Terre, you can easily catch a train up to Milan. Then you can catch a train east to Venice.
One problem with your itinerary is taking the train to Vienna from Venice. The Dolomites and Austrian Alps are in between these two great cities, and they're about 350 miles apart. The trip via ground travel is made by two train trips and a bus ride in between. It is a very difficult trip, and there are no budget airlines flying in between those cities. You might want to check the airfares on Austrian Airlines to see if it's affordable to go there.
I would prefer to go to Rome, Florence, stay 3-4 nights in Chianti doing day trips with a rental car. Then take the trains to Cinque Terre and later over to Venice. Unless you have any reason to go to Nice/Monaco and Vienna, you should save them for another trip.