The thing J got when he awakened was a watch given by the system (i can't really explain the 3 worlds because im confused atm about whoch worlds J used the watch to rewind along with lsy on when he used it as well.)
Yoon Ga-eul (the girl) was stuck because her soul was forced taken by the shard (she made it as a necklace, i think) she got from coming to the dungeon with J and yeseong where jung bin and lsy were.
As for the last question the "home dungeon" was specifically requested by J to yeseong from the 3rd world (where lsy died), the home dungeon was created with J's memories with lsy (yeseong also had warned J that the present world is his last and final chance.)
i hope these somehow answers your questions, i rly need to get back to reading the novel again :')
i forgot that, no the home dungeon does not exist in the rift because their souls are currently in a certain fragment/shard if im sure and I'm guessing its the 3rd world
Can some kind soul explain what's going on?
Overall I think I understood, but I feel like I lost some details. Especially that thing he said he got when he awakened, and why that girl was stuck there, why a "home dungeon" exists inside the rift? (Or not?). Idk