Your questions are too broad. "Must sees" and "off the beaten track"... in three countries!! Even if I attempted to answer the must sees, they would be MY must sees based on MY interests.

Even though you're apparently leaving in a matter of weeks, you need to spend some time reading guidebooks or websites to get a sense of what places/areas appeal to you and your group. Click on "Destinations" at the top of this page and start by considering the main sights in each country and/or main cities and any suggested itineraries.

You can't answer the car vs. public transportation question until you know where you're going. However, I will say Switzerland is easy to explore using only public trans.

Yes Jean, you are correct. I have actually changed my original itinerary and am thinking of flying into Split, renting a car and driving up to Lake Bled and Julian Alps, returning to Split and flying or ferrying to Venice. Spend a couple of days there, then visit Lake Como and Dolomites. Switzerland and Austria will remain on the bucket list for now.
Again, any and all suggestions are welcome. These are the very early stages of planning.

Do NOT pass up Lugano - I would visit there instead of Como (although, being Swiss, it will be more expensive). Lugano has Italian charm combined with Swiss cleanliness and elegance. Take the funicular to the top of San Salvatore for an amazing view. The shops along Via Nassa are amazing.

If you rent a car in Lugano and are up for a day trip, go up to Valle Verzasca (between Bellinzona and Locarno) and drive to Sonogno at the end of the valley. You will be amazed.