If you want to spend a fair amount of time in the Dolomites and range rather widely, I think a car would be helpful. However, I spent about a week in that area in 2015 and got around quite a bit by public transportation. I stayed in Bressanone/Brixen and Bolzano (valley cities; most folks prefer to stay up in the mountains in a place like Ortisei). There's a train line running from Verona up that valley, stopping in such places as Merano and Chiusa/Klausen. Then there are buses running from the major towns up into the mountains, plus of course alpine lifts to get even higher.
I think the Dolomites are a great place to go on a summer trip because once you get up to the mountain villages, the temperature is usually relatively pleasant; the temperature may spike briefly in the middle of the day (see link below), but it is a lot cooler early and late. On the other hand, down in the valley it can be oppressively hot at that time of year, and air-conditioned lodgings are highly recommended.
Many people go to the Dolomites to hike. There are lots of options, including high-meadow strolls that anyone can manage. If your focus would be on such activities, I'd recommend spending at least some of your nights up in the mountains. Of the valley cities Bolzano is the largest and has the most traditional sights, including the museum holding Otzi, the Iceman. It also has a really beautiful historic center. Brixen is smaller and also has an attractive historic area, but it's more of a walk from the train station and perhaps not as convenient for side trips.
As of 2015 a lot of the lodgings in the area (not as far south as Verona) were giving their guests transportation passes that covered local buses and trains; that's something to check on when you evaluate the rates at different hotels, etc. But with a car you work out some sort of loop trip, I assume, rather than being focuses on one area.
I can't address off-season weather, but you can review actual, historical, day-by-day weather statistics on the website timeanddate.com. The data goes back about ten years and is much more informative than monthly averages. The link goes to Ortisei's weather chart for July 2020. Note the sharp mid-day spikes. It wasn't that hot on my June 2015 day in Ortisei; perhaps I was lucky. That is why anecdotal comments about weather experiences are potentially misleading. It's really much better to look at 3 to 5 years' worth of actual data.
It's also critical to factor in the impact of altitude on temperature. This website explains the situation and says on a day without precipitation, the temperature will drop an estimated 5.4 degrees (F) for every 1000 feet of elevation. https://www.onthesnow.com/news/a/15157/does-elevation-affect-temperature. Ortisei is at 4000 feet.