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There is no acceptable/viable mechanism for a free electron to absorb or emit energy, without violating energy or momentum conservation. So its wavefunction cannot collapse into becoming a particle, right? How do 2 free electrons repel each other then?

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.

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.

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.

Basically, because the process doesn't simultaneously conserve energy and momentum; this is why we say that it's mediated by a virtual photon.

In more technical language, this means that the photons that are exchanged between two interacting electrons are allowed to be "off shell", where the "shell" is the relationship $E^2 = p^2c^2$ (for a massless particle). Real particles are required to be on-shell, but virtual particles are allowed to stray from that condition, at least by some amount.

In the Feynman-diagram description of the scattering between two electrons you often find diagrams where one electron emits a photon which is then absorbed by the other; this is a virtual photon and as such both the 'emission' and 'absorption' processes are exempt from these considerations.

As always, though, it bears repeating that Feynman diagrams are calculational tools, and none of the virtual particles that appear in those diagrams actually physically exist in any definable sense. The fact that the 'emission' and 'absorption' processes, as well as the virtual photon itself, seem to defy energy or momentum conservation is purely a quirk of the way in which we've chosen to interpret limited chunks of our calculation.