Kent,
After reading over your Itinerary, the first point to mention is that renting a car in one country and dropping in another usually comes with a hefty "drop fee". When entering Switzerland by car, you'd also have to ensure that you buy the Swiss Highway Tax Vignette (which probably won't be provided on a car rented in Italy). Hefty fines payable on the spot if you're caught without one!
For driving in Italy, each driver listed on the rental form will require the compulsory International Driver's License, which is used in conjunction with your home D.L. These are easily obtainable at any AAA / CAA office for a small fee and are valid for one year. It Italy you'll also have to be extremely vigilant to avoid the ZTL (limited traffic) areas which exist in most Italian towns and cities. Florence is heavily covered by automated ZTL cameras so don't drive there! Hefty fines for each pass through these! There are also highway speed cameras, tolls on the motorways, fines for driving in bus lanes and of course expensive fuel. You'll also need to be aware that parking your rental car with all your luggage inside while you're touring or whatever, comes with some risks! You may find your luggage missing when you return to the car!
You may find this website interesting - https://www.italybeyondtheobvious.com/the-italian-autostrada-tips-for-foreigners .
Using a combination of high speed trains and car rental for out-of-the-way locations would be a better strategy (IMO). The high speed trains travel at 300 km/h, so much faster than any car!
Regarding your last question, my personal opinion on the places you mentioned....
- Dolomites - definitely worth a visit - Ortisei is a big favourite with many here, but I quite enjoyed Castelrotto and the Alpi di Siusi area.
- Verona - worth a stop if you can fit it in - if you're only going to see the fictitious Romeo & Juliet balcony, I wouldn't bother.
- Lake Garda - not my favourite - I'd suggest either Lago di Como or Lago Maggiore instead.
- Bergamo - I'd probably skip that.
- Milano - unless there's something in particular you want to see, I'd skip that.
- Lake Como - yes, absolutely! The small town of Varenna in the mid-lake area is a big favourite here.
- Lugano - perhaps..... if there's something you want to see there - you could also look at Locarno, which is nicer (IMO).
- Something you've overlooked? I'm surprised that the Cinque Terre is not on the list - while it can be dreadfully overcrowded at times, many people want to see it at least once.
Where are you visiting in Switzerland?