Asylum is not accessible until it's shut down. You will be able to explore it like any other map. Not just during ghost hunt contract. There will be generic looking ghost roaming it. If it catch him he will wake up in the hospital.
Timeless event was something that happens after ghost hunting is done. For college level it was 2 hours after you begin it. For Junsa's house it would have been 4:30 AM. In Magic Ring, there's not such thing as timeless event during ghost contract. Contract will begin at 10 PM (if it's not 10PM) will continue until Masuo leaves the site.
You will be playing timeline 0.7, the current one. If you helped a Goddess, she will offer you to be with one of your previous lovers. Now depending on the lover their past selves will replace them. There will be changes to a few characters.
- 0.0-Rangiku- Aoi and Chiaki doesn't appear as they are kids. Hitomi & Rangiku attends Evergreen college with you.
- 0.2-Renko-You don't meet her until you befriend Shizuka.
- 0.4-Ryoko-Michiko becomes elder sister and Hinata younger. She also attends college with you in her place.
Rumiko does appear but you will find that out in Magic Ring.