It's likely a myth that Rasputin was sleeping with the czarina. (Although there's no proof either way.) He had a very close relationship with the czar and czarina because their son had hemophilia, and Rasputin's presence seemed to help his condition. It was a serious illness, and the parents were desperate for anything that could help. But the hemophilia was a secret so nobody outside the family knew why the czar and czarina kept Rasputin so close despite dangerous rumours that Rasputin was the one running Russian policy. The idea that Rasputin was sleeping with the czarina was a popular theory because Rasputin was a well-known horny motherfucker.
Oh yeah - the classic move of saving people from doctors! Everything I know about this is from the Revolutions podcast by Mike Duncan which suggested that Rasputin likely provided an emotionally-calming influence that probably helped.
The theory is that he stopped them from giving him aspirin, because aspirin exasperates the condition and was also something that they would have likely given him since it was considered a cure-all.
But what I do have are a very particular set of skills, skills I have acquired over a very long career. Skills that make me a nightmare for people like you. Specifically, I have a huge cock and I can't be killed.