How can two electrons repel if it's impossible for free electrons to absorb or emit energy?

It is true that the reactions $$e + \gamma \to e, \quad e \to e + \gamma$$ cannot occur without violating energy or momentum conservation. But that doesn't mean that electrons can't interact with anything! For example, scattering $$e + \gamma \to e + \gamma$$ is perfectly allowed. And a classical electromagnetic field is built out of many photons, so the interaction of an electron with such a field can be thought of as an interaction with many photons at once. There are plenty of ways a free electron can interact without violating energy or momentum conservation, so there's no problem here.

To resolve this paradox requires study of time dependent perturbation theory; solving Schrodinger's equation with a time dependent perturbation corresponding to the interaction time of two particles.

If you do this you arrive at the following conclusions:

A single free electron cannot absorb a free photon ( $e + \gamma \to e$ is not a valid interaction)

A single free electron cannot emit a free photon ( $e \to e + \gamma$ is not a valid interaction)

However, two electrons can scatter by exchange of energy ( $ e + e \to e + e$ is a valid interaction)

In this later case it is common to refer to this process being due to exchange of "a virtual photon" between the two electrons. But this is just a description of the calculation of time dependent perturbation theory.

Your first statement is false: energy can indeed be added at will to electrons by accelerating them with electrostatic charge distributions, as for example in the case of rapidly varying radio frequency (electromagnetic) fields. Neither energy nor momentum conservation is violated in this case. Search on SLAC for more details about this.

Your other questions are unclear. I recommend you do the search, read a bit, and return here if you have further questions.