The diaphram is supplied by the C3-5 nerve roots on each side. The side of paralysis should be the side of the nerve injury. There are syndromes (Parsonage Turner Syndrome-see website) that can cause nerve injury without nerve compression.
Coughing and vomiting without an associated nerve compression will not cause nerve pinching. It is rare but you can rupture a diaphram by those actions.
Donald Corenman, MD, DC is a highly-regarded spine surgeon, considered an expert in the area of neck and back pain. Trained as both a Medical Doctor and Doctor of Chiropractic, Dr. Corenman earned academic appointments as Clinical Assistant Professor and Assistant Professor of Orthopaedic Surgery at the University of Colorado Health Sciences Center, and his research on spine surgery and rehabilitation has resulted in the publication of multiple peer-reviewed articles and two books.