I'm going to say outdoor.
The "door" part doesn't really have any significance. No one would say camping under the open sky is an indoor activity, even if there's a fence with a door around the campsite.
I think it makes more sense for the deciding factor be whether you're in a controlled or uncontrolled environment. And while part of the cave might be controlled if there's an artificial entryway or home, that's not what you're there to see.
In Iran, gender reassignment is legal, and they'll even change the birth certificate to match, from what I learned a decade ago.
Homosexuality, however, is a capital offense, so many gay people are pressured to transition.
Some conservative societies seem to have the attitude that it's better to go from one role with rigid expectations to another than it is to fail to meet the expectations of your original role.