Once you use the item, Flapple will appear there till the end of the day.
Time manipulation via changing Switch clock will cause the raid to disappear.
You can only catch it once (once you catch that Flapple, the raid disappears.)
I am not sure, but I think you can invite people every time you attempt to catch it.
It saves right as you use the Crystal, so the clock starts ticking down from there.
It follows Switch time, and ends at midnight (the Crystal usage doesn't count 24 hours for you, purely depends on when midnight is.)