What we tend to do is have a base somewhere for a few days and explore that area then move on somewhere else and repeat. It's ideal as you can get unpacked, ditch the panniers and just ride without any unnecessary weight & bulk. Davos is a great place for a few nights as is Nauders in Austria with loads of passes to play on for many days.
Ditto, otherwise I find you spend loads of time packing/unpacking and also you can normally find a town with several good rides nearby.
Previously I've stayed in Briancon in France (near lots of Cols used in the Tour de France), Interlaken in Switzerland (can do the 3 amazing passes, Grimsel, Susten & Furka), Merano (Stelvio & Timmelsjoch passes) & Cortina D'Ampezzo in Italy (Dolomites).
Not done too much in the east of Switzerland or Austria, I imagine theres plenty of good roads to find.