Could try Graubuenden area of Switzerland. Basically from Chur in the North, East to the Austrian border, South to the Italian border (ok this includes Engadin region but it’s pretty cool too).
1- Tick.
2- Yup, loads of singletrack, trains post buses and ski lifts at your disposal. A while since I’ve been but ski resorts like Lenzerheide, Flims/Laax etc would be best bets for DH and bike park stuff, loads of bike tour routes (lift/train assisted) round all the ski areas though.
3- Climbing wise there’s quite a lot of sport/trad crags but can be a bit dispersed. VF too – some of which are “sporting” in spds with a bike on yor shoulder. Not too into climbing whilst there so no specifics I’m afraid. Alpinism wise I’m not too sure of what summer challenges would be good. Plenty in winter, decent rock and some spectacular (if not major) peaks so I assume there will be stuff to go at.
4- anywhere other than the main towns is mostly small, “traditional” villages, plenty of goats, cowherds, meadows… Some pretty sweet, quiet luxury places to stay if you’re into that (check out the spa in vals for example) and have the chf.
I’m sure you can find stuff in the S of Germany and Voralberg regions, but I like that area of Switzerland. I’d probably choose to travel round (the area is compact) rather than base myself in one place – maybe check out the biking around Bormio and Livigno which was supposed to be good(?) too. The graubuenden.ch english site is ok – the german one is more useful if you can read basics.
[edit – bails – sports climbing is climbing bolted routes (more or less), alpinism is mountaineering, vf is via ferrata – ladders and steps up exposed rock (usually) where it might be/ is sensible to clip into in-situ protection]