No, this stick is unusable for Switzerland, you need the Fire TV Stick | Basic Edition (international version), which is apparently not available in the Galxus offer, which is quite grotesque from Galaxus.
If you already have such a stick, you can still use the complete programme with a trick:
The stick must be logged into the Amazon account (otherwise nothing works anyway). Then install the Prime App and select the Fire TV entry from there via the "Play via..." function. Miraculously, the stick plays it anyway.

Yes, but like Netflix, the Prime Video offer is always limited to the Swiss offer, regardless of which device you use to watch.