Go to timeanddate.com and look at the "past weather" page - choose a city, then choose any month for the last 10 years and see what the actual weather conditions were. It will give you a much better idea of what may come your way than any anecdotal experiences from us here.
I've been to Rome in low season February and there were long lines for the Colosseum/Forum and the Vatican. I'd expect them to be longer a month later. And even longer in June. You can buy tickets in advance for most places.
The Secret Itineraries Tour is very interesting if you like history.
If you need to ask which one museum to visit in Florence, maybe Florence is not your best option. It is chock-full of Renaissance painting, sculpture and architecture. If you want charming towns, this is not where you should be. There are lots of other places between Venice and Rome that you could choose.
Are you aware of how long and complicated or expensive it is to Positano? From Florence you can get a highspeed train direct to Salerno. The journey takes 4 hours. Then you need to take a taxi for a 1.5-2 hour drive to Positano. Or you can take a bus to Amalfi (runs every 30-60 minutes, takes 1.25 hours), then change to a bus to Positano (runs every 3- or 60 minutes, takes about 45 minutes). OR via Sorrento - take a highspeed train 3 hours to Naples, then the local Circumvesuviana to Sorrento (runs every 30 minutes, takes 1-1.25 hours), then bus to Positano (about an hour, runs maybe twice an hour) or taxi from Naples (1.5 hours). . . . with luggage.
The only reasonable way I can see for you to visit Pompeii on the way to Rome (given the details in the answer to your last question) is to hire a private driver or a taxi from Positano to Pompeii. Then you still have to take the Circumvesuviana to Naples and get a train to Rome. You'll need to allow plenty of time for the transfer in Naples since the Circumvesuviana only runs about every 30 minutes and is often late.