I think it looks like a pretty solid itinerary. Most of the day trips you mention can either be done or not done once you get there and decide if you have enough time in the base city itself.
Milan - yes Lake Como is a place to 'relax' but I've done it as a day trip from Milan and really enjoyed it. I've also visited it on a trip where I spent a few days, which was also nice, but a day trip from Milan works fine. Zermatt is a bit further but still doable if you don't mind the time on the train. But do check the weather and don't bother if it's going to be cloudy (which it often is). It also tends to cloud up later in the day so if you do go, plan to leave Milan as early as possible.
Venice - For many people three nights would be a minimum to just see Venice itself. An easy 'half' day trip might make sense (Padua, Vicenza) but getting all the way to the Dolomites is stretching it. You would certainly need a car for this part and of course a car is useless in Venice. I would rethink this part. If the Dolomites are a high priority for you perhaps skip something else and plan on at least one night somewhere up there - not necessarily in one of the mountain villages but at least Bolzano or somewhere close.
Florence - while Pisa is an easy day trip from Florence, with only two nights you only have one day in Florence itself. I would skip Pisa this trip. Florence is really much more interesting.
Sorrento - yes that's a good base for the places you mention. With 4 nights you can do one day trip to Amalfi and Positano, another to Capri, and a third to Pompeii and Naples. You can see Vesuvius from Naples and Pompeii. If you mean you want to take the trek up to the rim then you are going to shortchange your time in Naples and Pompeii. Some people feel they need a full day for each of them but if you are pretty fast moving you can do both in one day. These three day trips use up your three full days in Sorrento so Sorrento itself you only get time in after you return from your day trips but that might be enough. I love Sorrento.
Rome - three nights is a minimum. Any less and you'll just feel frustrated. Even with three days don't try to see everything. For example, I personally feel that seeing the coliseum from the outside it the highlight so if time is short don't bother going in. The forum is more interesting. If you feel you need to see the Vactican museums make sure you arrange a tour or you'll spend the whole three days waiting on line (only exaggerating a little here). The best part of Rome is the streets and squares. Make sure you have time to just wander, don't run from one site to the next.
Here's my photos of all these places - https://andiamo.zenfolio.com/