I'd reverse the course. Also be aware that a car is totally unnecessary while in Venice, while in Florence, while in Rome, as well as the Cinque Terre
Fly to VCE
Venice x 3 nights (2 nights are too few especially considering it would be your first stop and you will still be jet legged the first day).
Rent a car upon leaving Venice (rental car offices at Piazzale Roma)
Drive to Dolomites
Dolomites (Ortisei?) x 4 nights.
From here I would drive to Tuscany. Staying in a countryside villa or in a smaller town will certainly require having a car, staying in Florence would be best without a car.
So you could stay in a place out of town and visit Florence in a couple of day trips from there, or you may split your stay and do the small towns first then move to Florence, return the car, stay in Florence a couple of nights, then finally proceed to Rome by train (a car is also not recommended while in Rome.
If you want to add the Cinque Terre into the midst, be aware the place is ultra crowded at that time of the year. In that case I would go there between the Dolomites and Tuscany. The car is not really needed in that area, but you might as well keep it because you will need it before (Dolomites) and after (Tuscany). Obviously the car will stay parked the whole time at the Cinque Terre. There are a few parking lots at the Cinque Terre (Monterosso has the most spaces), however be aware that accommodations will be tight in July/Aug, so you might decide to stay in a nearby town and take a quick train to the Cinque Terre as needed. Levanto could be a good choice, but also Bonassola or even as far west as Moneglia, which is very quaint yet just minutes away from Cinque Terre. They'll be crowded too, but nothing like Vernazza and the other CT villages.
Number of nights minimum:
Florence: needs at least a couple of days, so if you stay outside of it, know that two days will go to Florence alone.
Rome: needs 3 days to see the main sights, so 4 nights would be best.
Tuscany (besides Florence); Florence will take 2 days, so you need to add nights depending on how many other places you want to see. There are tons of places to visit. A lifetime in Tuscany would not be enough to see them all.
Cinque Terre area: that is up to you. A couple of nights is good.