In a situation where there is no device plugged in a normal house
electrical outlet, does it still run an electric current, if so then
how large?
Yes and no. Let me explain:
In practical terms when there is nothing plugged into the outlet there is voltage but there is no current. The current has no place to go so it doesn't flow.
In reality, however, there is always current flowing because even air conducts a minute amount of current. It's extremely small but it is present and with the proper equipment it can be measured. But generally this is disregarded due to its extremely small amount.
I would assume there is no electric current in this case (be it AC or
DC), because there is no charge distribution between the energy source
and the sockets, although some electrons probably move between these
two points.
What you have written here sounds technical but it makes no sense. In fact there is charge distribution between the energy source (i.e. a GENERATOR) and the socket otherwise there would be no voltage. As far as electrons moving, that's the definition of CURRENT. As electrons move through a wire they create current which is what you measure with an ammeter.
Yes, electrons DO move in the wires between your plugged in load and the power company's generator(s) but not as much as you might think. Electrons generally move at about 0.02 cm/sec. in a copper wire. So at 60 Hz the electric field driving them will reverse before they travel very far. You can think of them as just "wiggling" back and forth in the wire and never really traveling very far.
The electric field, however, moves much more quickly than the electrons and that's why electricity seems to travel instantaneously. In a wire the E-field travels at a significant percentage of "c" or the speed of light or 3x10^8 m/s. So while it's not instantaneous it's very fast.
So the INFLUENCE of the electric field from the generating station is felt very quickly even if the electrons themselves don't move all that fast.