Question

a.) If you increase the voltage of the battery, the electrons would move? Why?

b.) If you increase the resistance of the bulb, the electrons would move? Why?

Answer 1

Increasing the voltage applied to a circuit of a given resistance will increase the current flow. That flow is defined in electrons per second past a point. So increasing the voltage increases the speed of the electron flow.
The number of electrons free to flow is a constant for a material.
For Copper that is one electron per atom.

Current in a flow of charges, and in a conductor, it is the result of drift electrons jumping from atom to atom in random motion. The voltage across conductor ends creates an electric field in it, giving an effective direction(push) to this random movement.

When the field in increased, it naturally increases the number of electrons drifting this way under the “push" experienced. This is what causes increase in current when field is increased with an increased in voltage. Speed of individual electrons as such is not increased, since it is a random movement after all, but the current goes up in the process due to increased drift, which goes up.

Things are different in a vacuum tube, where electrons are physically in motion directly influenced by the electric field. Out of the electrons generated at cathode of these tubes, some more electrons will move and also at a higher field force. Of course there is a limit to the number of electrons so moving, because that is decided by free electrons available at cathode area.