He wasn't sexually mutilated, at least according to the coroner who did his autopsy report. I don't have the book immediately on hand but I'm getting this from Douglas Smith's excellent Rasputin: Faith, Power, and the Twilight of the Romanovs which I highly recommend if this is a subject that interests you. The book goes pretty in-depth on his murder and the findings of the autopsy. His genitals were apparently fully intact when he was brought into the morgue.

The assault is a lot harder to really comment on because I suppose no one really knows what happened that night except for the people who were there. Personally, I find it very unlikely. By all accounts Prince Yusupov found Rasputin to be personally repulsive. The idea that he was somehow sexually motivated in the murder of Rasputin has been bandied about before but there's not much to corroborate it other than he-said-she-said years after the fact. Yusupov was infamously bisexual and had male lovers, for sure, but there's no evidence to suggest he had ever made passes on Rasputin. The murder was politically motivated; the murderers were all noblemen who were unhappy with his influence over Nicholas II and his wife. Yusupov was married to Nicholas's niece and his main cohort, Dmitri Pavlovich, was Nicholas's first cousin, so their family reputations were both potentially at risk from Rasputin's actions. I think the sexual motive idea probably comes from twentieth century homophobia and a desire to make the story more titillating. For what's it worth I've never heard anything about Rasputin being sexually assaulted prior to his death before this article and I've not been able to find any other sources for this claim.

This article holds very little merit, as it generally repeats the fictional version of the events popularized by prince Yusupov, but not corroborated by any other sources. First and foremost, the autopsy report published in 1917 by doctor Dimitri Kosorotov, professor of Military Medical Academy, stated that Rasputin died due to several gunshot wounds, with the ultimate cause of death being an internal hemorrhage caused by the shot to an abdomen (what suggests that the forehead wound has been caused post-mortem, possibly to make sure Rasputin is dead). No other wounds save for numerous bruises were mentioned. Also, no signs of poisoning or drowning were found during autopsy and Kosorotov explicitly concluded that Rasputin was dead when he was thrown into river. Both Purishkevich and Yusupov claimed that they unsuccesfully tried to poison Rasputin, but according to Shishin and Nelipa, Yusupov claimed that although there were plans to use poison, they never resorted to this method having absolutely no experience in such matters.

The fact that Rasputin started to run away and had to be shot several times are very plausible. Despite what one might see in movies, gunshot wounds, especially from a small caliber weapon are almost never immediately fatal, and unless cerebrum, cerebellum or medulla is severely damaged, wounded human is usually capable of more or less coherent actions. Given that the weapon used by Yusupov was most likely a .32 ACP Savage 1907 pistol, it was possible that it took several bullets and a severe beating to kill Rasputin. So, instead of dramatic fight between conspirators and seemingly immortal Rasputin, the death of the latter was most likely what one would expect from a typical (although amateur) assassination, as corroborated by the aforementioned Kosorotov's report.

Moynahan, B., The Saint who Sinned. Da Capo Press, Boston 1999.

Nelipa, M., Killing Rasputin. The murder that ended the Russian Empire. WildBlue Press 2017 [although a research is decent, the style and especially the editing is amateur at best].

Shishin, O.A., Распутин. История преступления [Rasputin. History of the crime]. Eksmo, Moscow 2004.