Well, if your cockatiel's toenails are falling off, it probably has some kind of sickness. take him/her to the vet and see what he says.
It's not.
It means... you are going insane.
Because the kid is a freak and should see a doctor.
It's not unusual for a toenail to fall off and grow back. If an infection is caught and not treated it may result in death.
A dogs toenails do not fall off. They need to be trimmed or they can be accidentally torn off. Yes, they will continue to grow all the time.
Your toenails fall off
Yes, if you trim them off your toe first.
Yes it is normal
Too small shoes.
Peanut butter
use a planner to plan them off
Not much. Keep the area clean to minimize risk of infection and wait for the new nail to grow back; that is if you didn't damage the nail matrix during whatever event that caused you to lose your nail in the first place.