We have been to the Dolomites 5 times and are going back in September. I don’t think your trip looks too rushed. Think in terms of nights ( where you will sleep) rather than days.
A car is not needed in the Dolomites fin you choose one of the locations served by the local buses, such as Val Gardena. There are enough beautiful hikes right there to keep one busy for a week, so you won’t have trouble filling 3 days.
I would suggest booking roundtrip to Venice, assuming there are good flights from your location (most or all will have one connection on the way). Depending on the time you land, you might like to continue to Bolzano right away. From the airport you would take the bus the short distance to Mestre, where you will catch the train to Bolzano. Most will require a change at Verona, but the entire journey is only 3 hours. There are numerous good hotels within walking distance of the Bolzano station.
The next day you can take the bus from Bolzano to your Val Gardena village. Many here will recommend Ortesei, which offers lifts up both sides of the valley (Alpe di Siusi on one side and the Puez-Odle group on the other).. That is an excellent choice. We have always stayed in smaller Santa Cristina, the next village up the valley. After that is Selva, which also has its advocates. They are all good, so find a hotel you like at a price you can manage. Book 4 nights to have 3 full days for hiking.
If you prefer to rent. A car, you can do so at Bolzano, usually to be picked up at the airport.
After your hiking time, return to Bolzano and take the train to Florence. There are direct trains that make the journey in 3 hours. The current schedule has these departing at 13:12 and 15:12, and that likely will not change much, but check back in June to be sure. There are lots of other options with a change at Verona and/or Bologna.
After your Florence time, take the train to Venice to be in place for your flight home. I hope you have 3 nights left for Venice so you can spend two full days. There is a Jazz Festival I see on the calendar for Sept. 24-Oct. 1, but I don’t know that it will drive up hotel prices much.
I am suggesting this order of travel because it gives you the better chance of good weather in the Dolomites, and (maybe). lower crowds in Venice. And I am suggesting roundtrip to Venice rather than open-jaw into Venice and out from Florence because there are fewer flight choices from that airport back to the US. (When my son spent a year in Florence as a student, he used the Pisa airport rather than Florence for that reason).
If you do prefer to fly open jaw, into Venice and out from Florence/Pisa, then the order of travel would be Dolomites (keeping that first) then Venice then Florence.
