I can't understand why a free electron can't interact with a photon (because of a violation in the law of conservation of momentum) when the reflectivity of metals depends upon photons interacting with free electrons. This seems like a contradiction. Can someone one please explain this to me.
First you have to learn that you are asking about two types of electron states.
You are talking about a free electron.
A free electron is unbound in vacuum, away from any other particle or atom.
That free electron cannot absorb or emit photons.
Now the electron inside the metal, that electron is not free. It is still bound to the metal, it is free to move inside the metal, and that is why it is a conductor.
Now the high reflection of metals, it is the reason for metals (most of them) have silver color.
In metals this is due to electrons on d levels (as per QM) to move to s orbitals. This needs lot of energy (ultraviolet) and so visible light does not have enough energy to match this energy level and so photons with the wavelengths of visible light will not be absorbed but elastically scattered.
That is how reflection works, elastic, Rayleigh scattering, and that is how a mirror image is built, because that is the only way a photon keeps its energy and phase.
That is why it is silver, because it reflects all visible lights, it does not emit any of its own light, it does not have its own color.