As convoluted as the true ending is, I think I finally have a relatively clear understanding of what happened. Or at least I think I do, that's why I'm checking to see if I'm completely missing something.
So if my understanding is correct:
There was the first Nonary Game, held by a pharmaceutical company, trying to tap into the morphogenetic field/the Force/Harry Potter magic. At the end of that game, Akane gets tossed into an oven and forced to do a sudoku puzzle. She doesn't know how to do sudoku puzzles, but she's able to finish it by watching Junpei do it in the future. And because her survival of the game depends on something that hadn't happened yet, and potentially might not happen, Akane and Aoi arrange the second Nonary Game to make sure that Junpei is at the right place at the right time in order to prevent time and space from flipping itself inside out or whatever. Also, a mummy lady (maybe).
Does that cover most of the bases?