I don't think so ours certainly didn't.i took a travel kettle cups etc.there is a spar supermarket just down from the hotel for fresh milk and a fridge in all rooms .The spar also has reasonably priced beer and wine which we enjoyed on our balcony, plenty of room in the fridge for a few bottles.
To my knowledge they don't and most people take their own with them if they had it. Certainly not in our rooms.
Yes, our room had a small supply of tea/ coffee and kettle. As we travel a lot we always take a small travelling kettle and supply of tea, coffee & creamer powder to cover those occasions when these essentials (to the travelling Brits) are not supplied.
Get an answer from Sporthotel St. Anton staff and past guests.