There definitly a hierarchy of which feather takes how much time.
The stone fades 4 times before it disapears.
0 fades = you get a regain feather (hp/mp/tp)
1 fade = you get a rebirth feather (reraise3)
2 fades = you get revive feather (reraise1)
3 fades = you get fire feather (enfire)
4 fades = you get blaze feather (blaze spikes)
5 fades = mimeo stone disapears and you dont get anything
I tested this, because i got regain feather before anything else, without any fades.
2nd time i got up, with no fades, and it said "You cannot obtain another regain feather." so i waited, and eventually it said "the mimeo stone starts to fade away", and i clicked the cradle again, and i got rebirth feather.
i did this again, and after "you cannot obtain another rebirth feather" i got revive feather
So yes, you can hold all 5 feathers at the same time, after you use a feather (only tested with regain so far) you have to wait 2 hours before using another medicine.
There are 6 pops of luminent, scattered around the mountain. The 1st time that I did it, i went down again, and luminent was up right away. The other times it has taken a long time for it to repop. I suspect either its a random repop time, or that it pops at a certain time during the day, i tested 0:00, but it wasn't up.