In a metal wire, do protons flow in the opposite direction from the positive terminal, like electrons do from the negative terminal towards the positive terminal?
No. In a typical wire, say copper, the protons stay in the nuclei of their respective atoms. Protons do not contribute to the current flow in a wire.
You can see that there are many copper atoms structured in a crystal lattice, where neighboring atoms are bonded to each other by sharing electron valences. The outermost electrons in each atoms electron cloud are free to hop around so long as each atom of coppers charge remains balanced. I have always liked the golf-ball in a tube versus the water in a pipe analogy to describe current flow. So picture a tube filed with balls, the tube will represent the crystal lattice and the balls will represent the electrons inside the crystal. Now try to put anther ball into the tube. That ball will push the rest through until one pops out of the tube. This same mechanism is what causes current to flow in a conductor, as soon as an excess of electrons are introduced to one side of the conductor, provided there is a path for those electrons to flow, the electrons will flow through the cloud and exit the conductor to the rest of the circuit. Since the protons do not exist in a cloud structure, but rather in a rigid crystal, they are not mobile and remain stationary. Also think of Newtons Cradle.
Materials that do not have this same property are poor conductors, such as plastics or glass. Here is an article from wikipedia on electrical conductors