Can cervical spine root damage cause permanent paralysis in your entire body? From c1-c8? Can a pinched nerve in this area cause full body paralysis?

Not really. Spinal CORD runs from brain down spine encased in column of vertebral bones. Spinal NERVE exits between bones providing movement and sensation to a very specific body area. Damage to NERVE can produce loss of function and sensation only in a local area. Damage to CORD can produce loss of function and sensation below the level of the injury. Injury above C4 paralyses respiration - often fatal.
Nerve root or cord? Nerve root no, spinal cord yes a spinal cord injury at this level would spare arms but paralyze the rest including bowl and bladder and sensation if complete spinal cord injury a c8 nerve root injury would affect Just the triceps and hand intrinsic muscles.