So I did extensive research one what each person did after Junko's death, and I think I solved it. I did my best to fill any plot holes in. (Like how Junko got crushed but some still took some body parts?)
I'm sure that the thing used to crush Junko in Trigger Happy Havoc and DR1 Anime was kind of rusted and old, so maybe it didn't crush her completely. This would explain why some Remnants of Despair took specific body parts, and why the rest didn't take any at all.
Fuyuhiko clearly took Junko's eye, replacing his own. According to ongoing medical research, if a body part is transplanted into a living person from a dead person, it will not rot, due to now recieving the necessary vitamins and blood flow from the living recipient. There are conditions, however, that determine this fact. The eye will still rot if it is not transplanted into Fuyuhiko's socket correctly, or if Junko's eye is not compatible with Fuyuhiko's body. I'm not sure if the show, game, or mangas determine whether Fuyuhiko ends up keeping Junko's eye or not after the transplant, but I can only assume that he doesn't due to only ever being seen with an eyepatch afterwards. This means that it was not transplanted correctly, due to his lack of surgical knowledge, or his body was not compatible with Junko's eye.
Nagito took Junko's left arm, and this has the same explanation as Fuyuhiko's.
I assume that Teruteru, Sonia, and Mikan all killed their parents. Teruteru, I believe cooked and ate his mother, while Sonia killed her parents in order to have full control over her country in order to garauntee its demise. Mikan killed her parents due to abuse from family and friends, and I assume that her parents were part of this abuse.
Mikan also took Junko's ovaries, in order to bear her children. This has the same physical explanation as Nagito and Fuyuhiko's.
Kazuichi is seen attatching machine guns to cars and trucks, and committing mass murder with such, along with building mechanical robots to wreak havoc.
Akane would be the one that inevitably starved herself to near-death in order to feel despair, to her eternal love for food.
Nekomaru coached people against their will, eventually working themselves to the point of death by heart-attack, heat stroke, stroke, and anything else from heat, exhaustion, or stress.
Ibuki and Hiyoko took many lives at their concerts, commiting mass murder at each one of their performances.
Gundham used his Four Dark Devas of Destruction and other animals to wreak havoc as he enslaved the creatures.
As for who violated Junko's corpse... this all leads to Hajime... or Teruteru. So either Hajime Hinata/Izuru Kamakura or Teruteru Hanamura inevitably yet unjustifiably had sex with a corpse.
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You'll find out what Y/N did after Junko's death in the story.

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Against All Odds || Fuyuhiko Kuzuryu x Fem!Reader
FanfictionY/N Y/L/N, the Ultimate Poet, is the 17th student of this year's Hope's Peak Academy class. As the terror of this killing game sinks in, she takes an interest in one of the other students... Fuyuhiko Kuzuryu. There's just something about him... Some...