Are there any privacy respecting sites that give a psychological evaluation after answering some questions?

Not any that legitimate psychology bodies would approve, I don’t think.

I don't see a way to do this and maintain medical ethics, assuming you want this to be medically binding in any way. A psychological evaluation where they just say "well you have X" after five or six questions is a fantasy, and I don't see a way this could be done safely, sanely, and in a manner that respects medical privacy.