I actually remembered this question last night I need to wait to get online and when there's an event raid for me to test the last 2, but I can clarify the first 2 points at least.
1. I didn't test it with online, I tested locally, but what I did was I beat the raid, waited for the other switch to enter the catching animation, and close out of the game on the host switch. However, for online, I assume I have to wait a bit longer, like when the others exit the raid.
2. Since I found gimmighoul, I can verify that if you find the shiny, enter the raid and exit without defeating the shiny, then save, the raid (and shiny) will still be there for you to return to if you change the date to keep it the same day you found the raid. The raid will only disappear if the clock goes to a different date and you save, or if you defeat the raid and save.