I think I'd use tweezers to remove the red filaments after the flower had fully opened. Just a guess.

Still wondering when is the best time. Does it matter what color the stigmas are? I see them go from yellow to orange to red as the flower matures.

I do know that if I wait too long it becomes very difficult to remove it from the wilted flower.

Interesting question. I think I remember reading somewhere you just pluck out the threads when the flower opens and then let them dry on a paper plate or towel somewhere warm. When they are harvested commercially the whole flower is picked and then brought somewhere else to pluck, but I think that's more due to it being so tedious than anything else and not something anyone wants to do out in a windy field.