Taking your wish list in order:
Turin/Torino: we have not been, but this is my sister’s favorite place in Italy. They have been all over Italy from Sicily to Sardinia in 6 trips, plus she studied in Florence while in college. So naming a favorite carries a lot of weight with me. One thing you could do there is visit the amazing Forte di Fenestrelle, the so-called “Great Wall of the Alps.”
https://curiosity.com/topics/italys-fenestrelle-fortress-is-known-as-the-great-wall-of-the-alps-curiosity/
https://www.fortedifenestrelle.it/
I am going to try to find a way to add this to our own trip next July.
Aosta/Mont Blanc: you should consult with your friends regarding the availability of snow-free hiking in late May/early June. When we hiked the Tour du Mont Blanc in 2018, our guide mentioned that the early June trips she guides often have to be altered or detoured because of snow. We will be hiking with her in Gran Paradiso National Parc (just south of Turin) next summer, and she doesn’t want to go before the third week of July.
Milan & Varenna: I would put this at the end, giving you a day there before flying home. When you land, go straight to Varenna, as late May is a good time. We have stayed several times at Albergo Milano, in the large top-floor room in Casa Rossa, separate from the main hotel. This room is large and bright, with a huge balcony overlooking the lake, and a kitchen corner in case you wish to make a simple meal (breakfast at the main hotel is included, however). There is no elevator so you have to walk up to the 3rd of 4th floor, but you are hikers so should have no trouble with that.
There is an excellent cooking class you can do at Varenna, run by Chef Moreno at Il Caminetto, a restaurant above town. They will pick you up in Varenna and drive you there, and return you to Varenna or, as we did, to the Castel Vecchio where you can look around the ruins and maybe catch the falconry exhibit before walking back to town. His recipes and techniques are simple but very tasty——something we use at home.
Dolomites: if you want to hike, don’t stay in Bolzano. Stay in Castelrotto or better yet Val Gardena or up on Alpe di Siusi. Early June may or may not be early for completely snow-free hiking, but there should be at least some trails fully open. I see that Refugio Bolzano/Schlernhuette above Alpe di Siusi opens June 1, so that should be an indicator. But tha later you can put this stop, the better. Which brings me to . . .
Venice. This fits best after Turin and before the Dolomites. If you are going to spend 2 nights here (as you should), stay IN Venice as others have advised. It is false economy to stay elsewhere (like Mestre) to save a few euros. Venice is at her best in the morning and evening, and you will maximize your time and enjoyment by eliminating the commute from elsewhere and sleep on the island. Not all accommodations are expensive; you might consider extending your stay to 3 nights and getting an apartment through AirBnB or an agency (we rent from Venice Red House which is local and less expensive than the better-known Views on Venice and others).
Verona comes next, and would be on your way to the Dolomites if you travel by train. I would not skip Verona. It is lovely and full of history: the whole city is a UNESCO World Heritage site. Two nights are enough. You should be there before opera season starts, so hotels are less expensive.
I would skip Genoa and Cinque Terre and head straight to Florence from Verona. That gives you a total of 6 stops in your two weeks, which I do not consider excessive. (Turin, Varenna, Venice, Dolomites, Verona, Florence). Oh, I guess you have to add a night in Milan. . . . If you do feel the need to drop another stop, I guess I would suggest Verona. And shorten your time in Florence to 3 nights, not 4, which you don’t need if you skip the Cinque Terre.
