You choose the target for the triggered ability when the ability is put onto the stack. You choose whether or not Cryptoplasm becomes a copy of that creature when the ability resolves.

The copy effect lasts indefinitely. Often, it will last until it is overwritten by another copy effect (if it copies another creature on a future turn, perhaps.)

If the creature is an illegal target when the ability tries to resolve, it won’t resolve. Cryptoplasm won’t become a copy of that creature; it remains whatever it was before.

If another creature becomes a copy of Cryptoplasm, it will become a copy of whatever Cryptoplasm is currently copying (if anything), plus it will have Cryptoplasm’s triggered ability.