If I were to pickpocket all the energy cells out of a protecton, would it run out of power and shut down?
No. The energy cells are used as ammo for the Protectron's weapons only. It doesn't use them as a power source.
However, if you pickpocket their ammo, they will no longer be able to fire their weapons. This doesn't mean the amount of ammo they have defines how often they can fire, this is actually unlimited so long as they have at least one round.