In Europe in general, lodges aren’t really a thing, so there usually isn’t a place to stow a pair of shoes for the day. People tend to just walk to the lifts in their boots. Looking at the map, it looks like you’ll be a pretty short walk from the Rodelbahn. The Schlosskopf chair is closer as the crow flies, but it looks like you’d have to walk through a field to get there directly from your hotel.
The Rodelbahn is not a lift… it is a toboggan track!
i agree with the ”lodges” comment. There really is not the idea of a “base lodge” in Austria, as I understand them to be in USA. You might find lockers at a railway station, and certainly in ski shops, but the latter are reserved for customers hiring from that store.
I don’t know if you will be able to ski down from the Edelweiss. If there is decent snow, and the roads have not been gritted, then yes. You might be able to scoot down between buildings, but that I don’t know.
The walk is not that long, probably <10min.