There may be a good place for everything you want in Austria, but as someone who has taken teens several times to Switzerland, I'll weigh in on that.
Yes, Switzerland is somewhat more expensive than Austria, but if you get a place with a kitchen, the food expense comes down to what you'd spend at home.
We camped at www.campingjungfrau.swiss where you can rent a trailer or bungalow with a kitchen. From there you are centrally located and you can visit a cheese farm and even see cheese being made at Schiltalp. (The trouble with some farm stays is that you can be out in the country away from the action, whereas in the Jungfrau region you are close to Everything you are asking for, especially hiking.)
There are many "sleep in straw" farms throughout Switz. ( https://www.myfarm.ch/en ) and at least on in Grindelwald ( http://www.schlafimstroh-grindelwald.ch/ ) which isn't that far from a Rodelbahn at Pfingstegg and Trottibikes at First.
https://www.jungfrau.ch/en-gb/grindelwaldfirst/trottibike-scooter/
My 3 young teens loved the experience and even returned to the same trails as adults. You can see the trails and the Rodelbahn and maps at www.lauterbrunnenhiking.wordpress.com