They aren't among the "seven words you can't say on television."
People were once conscious of a vulgar meaning of "nuts" (testicles). "Nuts" still has that meaning, but nobody thinks of it when they say or hear "are you nuts?"
They are strong simply because of their meaning.
"Are you dumb?" is insulting, but no worse than "are you stupid?"
"Are you nuts?" is insulting, but no more so than "are you insane?"
They definitely can cause hurt feelings if people think you are serious. They are the kind of things you say about someone else who's not in the room and can't hear them.