1) there is enough variation in the Dolomites to stay there a whole summer. You'll need to stay for a few days given that the weather is pretty unstable. I always base myself in Cortina. In my opinion the Dolomites are way more beautiful than anything in Austria and most places in Switzerland.

2)I love Innsbruck, don't like Salzburg. I love Venice too (i definitely don't agree you have to keep out during the summer as most tourists leave early) but for a mountain trip I'd fly to Innsbruck. Having said this, Innsbruck is so badly connected that we always fly to Venice, Milano or Zurich. Zurich is a bit far away for the Dolomites but we have relatives there and there are some wonderful roads taking you to the Dolomites. If you have time a stop in St.Moritz or the smaller villages could be very nice.

3) I love Vienna (relatives there too) but not during the summer as it can get terribly hot, worse than Venice that at least has a nice sea breeze and lots of shadow. Vienna-Dolomites-(Venice)-Innsbruck would make sense (no fee for dropping in INN). Cortina-Vienna by car is quite boring unless you take the Grossglockner road.

Copenhagen is nice, but I'd spend my time in the mountains (add Jungfrau?) or doing the castles outside Paris (Versailles, Chantilly, Vaux-le-Vicomte, Saint-Germain-en-Laye, Fontainebleau...). Don't limit yourself to downtown Paris! But I prefer Swiss or Italian food. I like Danish bread though.

You can drive all cars into Italy except luxury cars.