Is the motion of proton in EM described by the Schroedinger equation? Does the usual Schroedinger equation describing the non-relativistic motion of electron in electromagnetic field also describes non-relativistic motion of proton? (Of course the values of charge and mass of electron should be replaced by those of proton).
 A: Not only protons, but also nuclei can be described by the Schrödinger equation, and in fact one routinely does so.
When can we do so?
Obviously in the non-relativistic limit, i.e., at low energies. One aspect of such a low energy description is that the internal structure of the proton/nucleus is inaccessible, i.e., we can neglect all the interactions except the electromagnetic ones. This imposes interesting limitations on the description of neutrons and other charged particles - although they still can couple to the electromagnetic field via their spin.
When do we actually do so?
When solving Schrödinger equations for atoms one starts with a system of particles including the electrons and the nucleus and then transforms to its center of mass. Since the nucleus is much heavier than electrons (even in a hydrogen atom), one often talks about electrons moving in a central field with a fixed origin. However, many books on quantum mechanics do present this derivation.
Another case when we do so is in applying the Born-Oppenheimer approximation for solids, separating the movement of heavy nuclei and the light electrons. Note that we then go on studying the quantum behavior of these nuclei, e.g., when quantizing the lattice vibrations (phonons), although the explicit Schrödinger equation is rarely written in this context.
