Is it normal to feel pain down my arms and legs after i sneeze?

No: This may be a sign of herniated discs in the neck. With the trauma and movement of the spine during a sneeze, the cervical rootlets may be being stressed. This might be worth getting checked by your pcp.

No it suggests that : There may be some compression of the nerve tissues within your neck or cervical region that could be due to a variety of reasons including a disc herniation or spinal stenosis (canal diameter too small for the spinal cord) or both. There may be other reasons so you should have a thorough evaluation by a neurologist.