When a photon interacts with an atom, three things can happen:
elastic scattering, the photon keeps its energy and phase, and changes angle, this is the case of a mirror, reflection
inelastic scattering, the photon gives part of its energy to the atom and changes angle, this happens when infrared light transfers kinetic energy to the vibrational motion of molecules (heats up)
absorption, the photon gives all its energy to the atom, and the absorbing electron moves to a higher energy level as per QM
In your case, reflection, is elastic scattering, and this is the only way to keep the energy level of photons, and to build a mirror image.
Of course this is assuming a stationary reflecting surface (relative to the observer), mirror.
As the other answers say, when the reflecting surface is traveling towards or away from the observer, the energy level of the photons can change.
It is very important to talk about specular reflection, like a mirror, where the relative angle of the photons is kept too. And differentiate it form diffuse relfection, where the relative angle of the photons is not kept.
You are asking whether the momentum of the photons can be transferred to the surface. Yes, photons can exert pressure on the surface of the mirror.
Please see here:
Can something without mass exert a force?