Venice to Rome

Regarding car rentals, search here on the Forum for all the horror stories regarding very expensive traffic tickets arriving at your home 6+ months after the trip. Rental cars, gasoline, and toll roads are all expensive in Italy now, not to mention the ZTL zone/ tickets many folks didn't even realized they had violated.
Take the train!

With ten days, I would plan on no more than three destinations. You definitely do not have time for all on your list. You would only benefit from a car in rural Tuscany.

ETA: First, I would decide how much time you want/need in Venice and Rome, as those are a given based on your flights, even if you intended them as brief stops. Once you have that sorted, you can more easily choose the in-between. There would be absolutely nothing wrong with just choosing Venice and Rome! I am assuming you intended to spend at least a few days in each, hence one stop in the middle. Florence would be an obvious choice, but you can choose whatever you like.

The time needed to pack up, check out of the hotel, actually travel to your next destination, and check in at the new hotel will consume at least a half a day everytime you relocate. As you have surmised, this will be too much moving about to really enjoy the trip. Perhaps you should pare down the places you go to the two most important places. Definitely not more than three.

Arrival day and departure day do not count as your time will be taken up with immigration, getting to hotel etc., same for departure day you need to be at airport 3 at least 3 hours before you leave and if it’s a morning departure it could be early. So that leaves you with eight full days. Easiest to drop would be CT. Four nights in Venice 10/8 - 10/12, 2 nights Bellagio 10/13-10/14 and 4 nights Rome 10/15-10/18. You need to allow for travel time Venice to Bellagio and Bellagio to Rome. There are two websites Ron in Rome and The Man in Seat 61 which will give great advice On traveling in Italy, especially by train. Also the Trenitalia website for up to date train info…just make sure you use the Italian names for towns,Roma for Rome, for example. Train is the way to go….cheaper, easier on the nerves. The schedules for trains in October won’t be up yet, but they don’t change much. The earlier you book the high speed trains the cheaper…..

Use the 'itinerary' here for a guide to what might be doable in you time frame https://www.ricksteves.com/tours/italy/venice-florence-rome

Trying to do 4 locations in your time frame is mostly travel with very little visiting.

Given you short time frame I would not be inclined to do two 'water' locales.

You only have 10 nights.
3 Venice
3 Florence
4 Rome
Transfer by train

You can use Florence as a base to visit some Tuscan towns (e.g. Siena or San Gimignano both served by bus from Florence), however in that case you may want to borrow one night from Rome or Venice and add it to Florence.

Tara, you have 10 nights/9.5 days to work with, and the .5 one - which is your arrival in Venice - could be a jet-lagged fog. For all the reasons already provided above, yes, 5 locations are way too many for the amount of time you have. You don't want to spend too much of it packing, unpacking, changing hotels, dealing with train stations and sitting on transport, right?

So I'd get down to no more than 3. What those should be depends on your interests: why did you choose them, and which of them do you most want to spend time on? Venice and Rome are givens, as you're flying into one and out of the other, so you'll cut two of Lake Como, Cinque Terre or Tuscany.

Personally, I don't think renting a car for a first trip is a good idea. You definitely don't want or need one for Venice or Rome, and you could easily travel between Venice, Rome and your 3rd choice via public transit. Now, if you choose "Tuscany" you don't want to locate in the countryside without a car. Florence (capitol of Tuscany) makes a nice base for day trips to Siena, Lucca and some other locations by bus or train so that's an option. You might also hire a driver for a day exploring the countryside. Anyway, It's quicker to get to from Venice than it would be to the CT.

Venice: 3 nights/2.5 days
Florence: 3 nights/2.5 days (including 1 day trip outside of the city)
Rome: 4 nights/3.5 days

But if want you want to see in Rome is minimal, you could steal one of those nights, add it to Florence, and use it for a 2nd day trip. These are just suggestions so if you want to swap out Florence for Lake Como or the CT, that's entirely up to you. IMHO, you're just going to experience more with less stress if you reduce the moving around.