Yes, yes, yes, and NO. Naples and Positano will be even hotter than Rome, and Positano more crowded. And they are in the opposite direction from the Dolomites.
Between crowds and possible strikes, train travel in Italy is stressful enough as it is, so I would not add 2 more train trips. And I would not take time away from the Dolomites; I like a minimum of 4 nights there and 5 is even better.
To deal with crowded trains, I suggest booking First Class on high speed Trenitalia or Italo trains only, not regional trains. Then you will have reserved seats. I prefer Italo; some say they are less prone to strikes than the state-run Trenitalia, but I do not know if that is true. I do like their trains and their prices. Book early (3 months or so in advance) and you will get a good price in First Class.
Italo does not serve Bolzano but they go as far as Verona, and personally I would make an overnight stop there anyway, to break up the long journey. The Roman amphitheater in Verona, still in use for concerts and opera, is nice to contrast with the ruined one in Rome.
Then continue to Bolzano, changing there to the bus into Val Gardena. You can stay in any of several villages—-Ortesei is a favorite here, but we have only stayed in the next village, Santa Cristina so I cannot compare them. Hotel Interski is a nice family-owned hotel with a swimming pool, which should appeal to your child (and you as well).
https://www.hotel-interski.com/en/information/index/1-0.html
But I am sure you will find hotels to like in Ortesei as well.
Those are my tips. . . .