Photon hits an electron perpendicular to its velocity, Relativity and Work?

Here is the Feynman diagram for Compton scattering.

Using it one can calculate the crossection of photon electron scattering. One has to realize that both electrons and photons are quantum mechanical entities, and their interaction cannot be deterministic event per event, the way you imagine. The only predictions are distributions of many interactions that can be compared to the result of computing the diagram above. i.e. $Ψ^*Ψ$ , where $Ψ$ is the wavefunction of the specific system that is interacting.

The mathematics is such that special relativity holds and energy and momentum , angular momentum and quantum number conservation where relevant, event per event. Just it is not possible to set up deterministic conditions event per event. Only distributions can be compared with experiments.

Your 1.2.3. are in the framework of classical physics, and quantum mechanics had to be invented because classical physics does not work in the microworld

1: First the photon expends no energy as it pushes the electron that is not moving in the direction of the push.

2: Then the photon expends some energy as it pushes the electron that is now moving in the direction of the push.

Now maybe 1 and 2 happen in zero time, simultaneously, or whatever. I mean, if my description is too simple, the reader may add some fancy stuff according to his taste.

Anyway, the definition of Compton-scattering is that the thing that the photon scatters from, like electron, is a low-mass object that the photon can set in motion.

While the definition of Thomson-scattering is that the thing that the photon scatters from, like electron, is a high-mass object that the photon can not set in motion.