If I wanna copy a pet in the tome, do I have to come back every day? I noticed when I hatch w/ that pet it goes away
When we hatch with a kiosk pet they go onto a 1 hour cool-down. Our own timer takes 12-24 hours to reset naturally but a kiosk pet only takes an hour. If you want to hatch again you can go back in 1 hour and hatch again, but you will need to reset your timer or move the pet to another wizard (remember this drains energy to take it out of the shared bank).
There is a one hour cool down for pets to be hatched with again in the kiosk, but you will also have a cooldown like what Victoria said, the pets in the kiosk will last 2 weeks before being removed to let more people enter pets, or if they reach 10 hatches. Copying the pet itself will take around 10-20 hatches on average, sometimes as low as 5. Good luck on getting your pet complete!