Yes, in general plane waves are only solutions of the Schrodinger equation in regions where the potential is constant. The further imposition of boundary conditions when the potential is a series of steps generally restricts these states further (unlike in the free space case where the potential is the same constant everywhere). You can understand this quite physically as follows. The plane waves are states of definite momenta, but you know that in the presence of a non-constant potential, the
momentum will change, therefore these states can't have definite energy (they can't be energy eigenstates) when the potential varies with position.
The point about the phase velocity versus group velocity bit is simply that the group velocity is what corresponds most directly with what you are familiar with from classical mechanics.
