So. In the end, can someone tell the link between electrons, current and energy flow yet?
Okay, I'll make myself the laughing stock of everybody here, voluntarily, and will bite.
Current is defined as the net rate of flow of electric charge through a surface or into a control volume. Electrons are one possibility for carrying such charge. There are many different forms of energy, of which "electrical energy" –– really, easily accessible electric charges (batteries, accumulators, capacitors) or currents (DC or AC, like the mains current in your home) –– is just one form. All transfers of energy can be considered energy flow.
When sufficiently isolated conductors are used, the current flows in the conductor. For alternating currents, the current flows mostly on the surface of the conductor.
If the conductor becomes sufficiently hot, or the voltage between the conductor and a nearby other conductor becomes high enough, you can get an arc of electrons –– even if in vacuum. This can burn through the isolating material, and in a gaseous atmosphere, create a plasma arc. This, too, has useful properties.
Because moving charges always generate an electromagnetic field, it is also possible to
couple another conductor inductively (via the magnetic field) or capacitively (via the electric field). (The high potential difference between the conductor and a nearby isolated conductor is one way this coupling can occur; capacitively.) Although the field interactions then carry energy from one conductor to the other –– very much inducing useful current or voltage in the other conductor ––, we do not usually call this kind of energy flow "current", because the interaction is through photons, non-charged particles.
(One exception is with coupled inductors, such as in transformers. This is because current induces a magnetic field which in turn induces a current, and it is just easier for humans to say that "current is transferred", because saying "current in A generates a magnetic field that couples to B inducing a current". But, in a strictly physical sense, the energy transfer is mediated by photons there, and there is no "current flow" between the two.)
How much of the overall energy is carried by the conductor, and how much by electromagnetic field interactions, depends completely on the topology: what kind of curve the conductor forms, and whether there are other conductors nearby so that they can couple to the EM field generated by the current flowing in this conductor.
The question posed in the thread title, "Don't electrons push each other", is a complex one.
The simple answer is that
"That's a wrong question, because they interact
. 'Push' is a completely wrong concept here.".
The complicated answer is quantum mechanics behind
particle-wave duality. In essence, electrons are both fields and particles. When we look at interactions like the details of how electrons move in a conductor, we really need to look at the
fields instead of considering them as particles, because electrons exhibit almost completely field-like behaviour there, and they interact in rather unintuitive ways. Their behaviour has just about nothing to do with the marbles most people think about when they think about electrons; instead, they are
delocalized, like smeared over a possibly very large volume. The field of physics involved in this is
electrodynamics, or if we get to high energies or really fast phenomena,
quantum electrodynamics.
In a way, if we look at electrons bound to atoms or to a lattice like most metals, interacting electrons do not just 'push' but 'pull' and even 'twist' each other, depending on their quantum properties. (The last depends on the orbital magnetic dipole moments of the interacting electrons, and is often neglected.)
For example, the interaction between iron (Fe) and chromium (Cr) atoms in for example stainless steel, producing its corrosion resistance, involves these "magnetic" 'twisting' interactions between electrons.
But even that is just a crude analog, and probably makes any physicists reading this laugh at me for trying to describe it this way... It is one of those things that really is logical and rational, not "magic" at all, but is just not easily intuitively understood, because normal human-scale world does not contain suitable analogies at all.