@Rockdrop
My hot take on it is that the cat locks both his soul and Satomi's in limbo so he could salvage her. I think time moved differently in the limbo dimension, though, so it didn't actually take ten years of real time. Pretty sure the siren of the end was actually the ambulance siren, so that hints that time moves quicker there than in the outside world.
I believe that Chapter 1 is chronologically the last chapter to happen, taking place after the loop has been broken. You can see that in the end of that chapter, the cat basically gave her the choice to either die with the world (heaven's door) or live to do good (normal door). Loop Satomi is a bit bonkers and would see the world die with her (maybe due to her mother's influence), so I believe she'd choose to drop dead there. Her choosing to live means that the cat succeeded in reforming her, thus saving her.
I'm also pretty sure that the cat died from the accident, since he didn't follow Satomi exiting the door. He sacrificed his life to save her, just like how his partner sacrificed his to save the world.