I will address:
My question - Does not photon, which is supposed to be quantum of electro-magnetic field, interact with an electron "electromagnetically"?
A photon and an electron are elementary particles, quantum mecanical entities. Probabilities of interaction in quantum mechanics are calculated from the wave functions of the system in QED, using Feynman diagrams to get at the necessary integrals. . The "electromagnetically" resides in the coupling constants at the vertices, which is the electromagnetic one.
look at the lower diagram. A photon interacts with an electron, a virtual electron is the internal line, and a photon and electron leave as real particles. The "electromagnetically" resides in the couplings at the vertices, and the way the quantum numbers are conserved.
For example, why is that photon approaching an electron feels repulsive (and moves away from the electron) or attracted?
At the quantum mechanical level there is only a "scattering", angles and momenta can be calculated as probabilities of interaction. There is a probability of back scattering , but it is not a matter of attraction and repulsion.
Why is that these interactions are always characterized by transfer of energy/momentum? Moreover, what is photon 'hitting' the electron?
A photon "hitting" a particle means there exists a vertex at the Feynman diagram which has a non zero coupling, in this case of interacting with the electron.
Shouldn't that interaction be understood as "virtual photons" exchange between real photon and electron?
The interaction is at the vertices. It is the electron that becomes virtual in this diagram, i.e. it is off mass shell but still has the electron quantum numbers. It is off mass shell because it has the four momentum of the sum of the incoming two particles and is under an integral for the calculation. The mass of the electron enters in the denominator of the propagator that describes the internal line. To get the distributions, i.e. the probability function , one has to integrate the diagram for the variables of interest.