Does a spinal cord injury always mean paralysis?

No. Fortunately, not all spinal cord injuries cause paralysis. Milder versions of the injury may simply contuse the cord and may result in little to no neurologic deficits. Other more severe injuries may severley bruise or sever the cord causing a spectrum of neurologic deficits based on location and type of injury. Full recovery is less likely with severe injuries.
No. No-- spinal cord injury can be complete (paralysis), or incomplete (partial paralysis). Incomplete injuries can vary, with some near-complete paralysis to those who have full strength but only lingering numbness or simply abnormal reflexes.