They are both referring to a concept of a different dimension coming out of Einstein's theories.
No. The science-fiction concept of "hyperspace" has nothing to do with Einstein's work. Einstein only works with the normal three dimensions of space, and one of time - he merely allows you to *bend* them.
Some people believe that it is the set of dimensions that wormholes are moving through.
They would be incorrect, though we can forgive them that misconception, as it is common. Einsteinian wormholes do not need a higher-dimensional space, though Einstein does not preclude such from existing. Einstein makes *no* predictions about the qualities of higher-dimensional spaces, much less any that would tell us whether we could steer a ship through them once there.