1) Can a toilet drain be used as a sauna drain?
Yes, but we would need to see firsthand the piping arrangement you now have and what your sauna will require.
2) Will the water evaporate from the p-trap if not used
frequently?
Yes it will.
3) Is there another type of trap that prevents sewer
gas from entering, but doesn’t use water?
Not that I’m aware of. However, you could install a trap primer which will add water to the trap each time a fixture on that particular waterline the primer is installed on is used.