Does the plate feel an unbalanced force towards any direction?
Yes. The change of EM momentum on the reflective side is twice the change of EM momentum on the absorptive side. So the radiation pressure is twice as large on the reflective side, leading to a net force pointing from the reflective to the absorptive side.
Doesn’t the answer to the previous question (yes or no) violates either the first law of thermodynamics or Newton’s third law?
Of course not. Thermodynamics is irrelevant here since the radiation is not thermal. There is no temperature associated with monochromatic radiation, so nothing in thermodynamics prevents it from doing work. Moreover, the energy removed from the radiation is proven equal to the energy gained by the matter.
A more interesting question would have been to make the absorptive side a black body and to irradiate both sides with thermal radiation at temperature T. This will provide a force, as before, but now the force will depend on the difference in temperature between the plate and the radiation. The force will go to zero as the temperatures become equal. In other words, it is a heat engine.