A bit of history re the OP's query about 'do intimacy' (or 'do' almost any activity)...
This is a rather recent American colloquialism and well known. I believe it originated from a gag line in a television series or a movie. (Could it have been the TV show Seinfeld, perhaps??)
A man is interviewing a woman for the job of housekeeper and is telling her the various tasks she would be required to do for him if hired, to which the woman responds,
"Oh yes, sir, of course I'll do your dishes, clean your toilets, [etc, etc]," ...then adds forcefully, "But I don't do windows."
The man's –and the audience's– surprise at this one possibly never even mentioned chore being singled out as being simply beyond one's dignity and too much to ask for became a popular joke and an established American colloquialism: A catchy or cheeky way of inquiring or responding to someone's query. It's commonly used in the negative as a response to practically any kind of request or invitation for someone to do something:
"I don't do church."
"In case you haven't noticed, I don't do calm and peaceful!!"
"Jane doesn't do sushi."
–A similar cheeky way of responding is, "Sorry. X is not part of my repertoire."