--if I stay in Feldkirch, is it convenient to ski in Arosa or Flims?
It's possible. I recommend to drive to Churwalden, if you want to go to Arosa. It's the same ski area, but you are 30 minutes faster.
--I like piste skiing. Are Flims, Arosa, Flumserberg, or Toggenburg any better than Damuls and Montafon/Silvretta?
Flims and Arosa are much bigger ski areas then the others. Depending on what you like more, they can be better.
--Finally, if I drive into CH but stay off the motorways, do I need a sticker like Austria requires (and is it even possible to do that?)
No, you don't need any sticker. The sticker is only necessary, when you want to use the motorways. All destinations can be reached without motorways. Google Maps has the possibility to show you the way.
I am not sure whether you can reach the areas mentioned without the sticker.
Warum sollte man Skigebiete nicht ohne Autobahn erreichen? Es hat/muss(?) ja immer noch eine Alternativstrasse haben.
Skitage 16/17: 19 (6×Adelboden-Lenk, 5×Savognin, je 2×Obertoggenburg, Andermatt-Sedrun, Davos-Parsenn, je 1×Arosa-Lenzerheide, Flumserberg)
Skitage 15/16: 17, Skitage 14/15: 20, Skitage 13/14: 19, Skitage 12/13: 19, Skitage 11/12: 14, Skitage 10/11: 16, Skitage 09/10: 13