In mid-June 2011, we picked up a car in Stuttgart and spent the next 2 weeks driving around and in the Alps.
We did stay at the highest hotel on Grossglockner. It was raining when we got there. There was about 10" of snow the next morning. We were not able to leave until about noon when the shop operator came up in her 2WD car and said things would be okay if we went slowly. Even the mega 4WD owners waited until then. The guys on motorcycles waited till a little later.
That incident made us decide not to drive Stelvio Pass in Italy, but we had lots of fun taking back roads and a few lower passes the rest of the time.
We drove through Germany, Austria, Italy, Switzerland and France and returned the car just over the border from Basel in Lorrach. For different amounts of time depending on what we were doing there, we stayed in Fussen, Berchtesgaden, Grossglockner, Merano, Varenna (Lake Como), Zermatt, Annecy and Basel (the last night before we flew home).
If you look up those places, it will be obvious that all are not exactly Alpine. Our trip was more about driving and gawking than hiking or walking.
We both liked the Dolomites the most and enjoyed staying in Annecy and at Lake Como the best. You will find your own favorites.
One of my favorite parts of that trip was flying OVER the Alps from Florence to Stuttgart. We were in a turbo prop plane. It was clear and we could look straight down at the peaks. We weren't all that high above them.
Wherever you decide to go and whatever you decide to do, you will have a great time there.