Family with kids - 10 days in june

1.-2.-3. You don't really go to Positano for the beaches but just for the setting. If you are interested in the Amalfi Coast, Minori and Maiori have sand beaches. I would think kids would find Naples and Pompeii much more interesting. Also look at Ischia.
Of course Sicily is interesting, but I would not want to fly in the middle of a short vacation when there are so many places on the mainland. Venice is just a short train ride away, so I would not discount it so easily. I don't know how interesting kids would find Renaissance art in Florence, but Venice is, well, Venice. I always though Puglia would be interesting to kids also, but a week might not be sufficient.
4. It depends on the itinerary. Italy has a fantastic train system, it is when you start going to villages, rural places that a car comes into play.
5. Once you get closer to a plan, google for festivals and such in each place.

Venice is one of the most unique cities in the world , and it is very different from Rome. I think the kids will love riding the vaporetos [water buses] and exploring the back streets, eating gelato, visiting the surrounding islands. When we were there a few years ago, we used ROW Venice, which gave rowing lessons to small groups, 90 Euros total, for 90 minutes. It takes you on the back canals, and I think the kids would have a great time- it's harder than it looks. There is also a beach on Lido, but I haven't been there, so not sure the quality, however, European beaches generally are not like US beaches in sand quality.
Can you fly into Rome and out of Milan or Venice- or the opposite?
Have you considered Lake Como?
Safe travels!

We took our kids a few years ago and it was a great trip. Here's my thoughts on a few of your questions.

1 - I can't speak to Positano or Amalfi, but we spent time in Ischia and are kids still talk about wanting to go back. The beaches and water there were fantastic. There's the castle to explore and Mount Empomeo to climb. That said, we were there early-mid September. I suspect water temps late June are still swimable, but perhaps cooler than we experienced.
2 - I don't have experience with Sicily, but that is quite a ways away to add to a Rome trip. I would think precious time would be far better to spent in the northern areas.
3 - See above regarding Ischia. Naples was also cool for a few days. We visited Mt Vesuvius and Ercolano, as opposed to Pompei. Ercolano was more the right size and crowd level for our family. I think the family's favorite thing in Naples though was the underground tour off of Tribunali and Gregorio Armeno. Very cool experience.
4 - We have always done trains and haven't regretted it. I imagine we'll get to a point where we've seen enough of the train accessible places and want to venture out by car, but we aren't there yet. So if you haven't seen much of Italy, I personally wouldn't recommend a car.

I'll add that our famliy LOVED Venice as well. Plenty to see and do. And I didn't think Florence and Venice were simply extensions of Rome at all.

We took our young adult kids to Lake Como which everyone loved. There is hiking there too and gardens which we enjoyed. It is fun to take ferries all over. We were there in August and went swimming.

We also went to Venice with them and it is nothing like Rome. There is a beach on Lido (vaporetto ride away) that is better in terms of sand quality than Lake Como which is pretty pebbly but Lake Como has views with mountains that are gorgeous. My kids spent an afternoon at the beach--we are from Florida and did not.

We have on a previous trip been to Positano and I would not go there with children. Mostly expensive shops and not much else to do. A ferry ride down the coast is beautiful though.

We have never rented a car in Italy.

For a nice beach trip, you could consider the Cilento coast, which is just south of Naples & Amalfi, and way less touristed than Amalfi. From there, it's easy to visit Paestum or take a ferry to Positano. We stayed in Castellabate 11 years ago, with our kids who were 16, 12 & 5 at the time & they loved it. The town is a medieval maze of streets with only Italian tourists. It would probably be difficult to do without a car, however.
This summer we're stopping in a different Cilento town, Pisciotta, on our way from Rome to Calabria. I haven't been there, but I hear it's also lovely. Marina di Camerota is another Cilento beach town that's been recommended.
If you decide to stay in Rome, you can take a bus to San Felice Circeo for a nice day trip to a beach. You could also take a day trip to Bomarzo, which is a fantastical garden full of bizarre monster sculptures & structures - fun for kids.