Do electrons actually flow when a voltage is applied?
Thinking about current in terms of electrons moving around is the start down a path to a poor mental model of how electricity works. Here are just a few things wrong with it:


Electrons are but one of many charge carriers. Any ion is also a charge carrier.
The protons that balance the electrons are just as important. If you had just electrons, then all the electrons in the universe would be repelled from each other and shoot out into the universe.
Electrons have negative charge, and you will confuse yourself for no good reason at all thinking about how they flow from negative to positive. It doesn't actually matter at all.
Electrons are actually swarming around in all random directions all the time, and their motion due to current is minuscule, in comparison.


The important thing is this: charge carriers (electrons being one of such) can be used to transmit an electromotive force (usually called just voltage). This is a pretty ordinary concept, really. You can push one end of a rod, and transmit a mechanical force to the other end of the rod. Does the rod move, when do you this? Well, maybe, but there are two things happening here:


force is transmitted through the rod, as waves propagating at the speed of sound in that material
if and only if we also transmit power, the rod moves, at most cases at a much slower speed


The difference is obvious for a rod, but since we can't see electric charge, the difference is not obvious.

So, your question was: Do electrons actually flow when a voltage is applied? Strictly speaking, the answer is maybe, and it depends on what you mean by flow. It's similar to the question, does a rope move when you pull it? Well, if it's attached to a balloon, it might move a lot. If it's attached to a brick wall, it might not move at all.

The motion of charge carriers (like electrons) is current. If we have a current, then there is a net motion of charge carriers. Really they are swarming all over, much as the individual water molecules are swarming around in a pipe, even if there's no net flow. The current describes the average movement. In the case of DC current, the average motion is in a circle.

How the individual charge carriers interact to accomplish this is complicated, and it's really a physics question, not an electronics question. However, I'd suggest you check out this MIT tutorial on fields.