What could be the cause of severe pain in the right lower back / hip area when walking, sitting coughing moving a certain way ?

Low back pain: Radicular back pain, which is typically sharp and radiates from the lumbar spine to a leg. Coughing, sneezing, or voluntary contraction of abdominal muscles (lifting heavy objects or straining at stool) may elicit the radiating pain.

Disc: Most likely herniated disc with nerve root irritation. Possible spur/stenosis.

Hip or spine: It could be hip arthritis or a pinched nerve in the lower back.