if a magnet is moved in and out of a coil of wire, electricity begins to flow through the coil?
yes, this is called alternating current or AC, which was invented by Nikola Tesla, this allows the electrons to move back and forth in the circuit.
No. Moving the magnet in the coil only produces an electromotive force (voltage) between the ends of the coil. If there's just a space between the ends, and nothing to carry electric current between them, then nothing flows.