Electrons always exist in the circuit as part of the atoms and molecules that make up the circuit. The electrical energy that is delivered is the result of the electrons moving through the circuit. Turn off the pump (i.e. disconnect the battery), and the electrons stop moving through the circuit.

Where do electrons eventually go and how?

As electrons move through the electron transport chain, they go from a higher to a lower energy level and are ultimately passed to oxygen (forming water).

Where do free electrons go?

Metals atoms have loose electrons in the outer shells, which form a 'sea' of delocalised or free negative charge around the close-packed positive ions. These loose electrons are called free electrons. They can move freely throughout the metallic structure.