Does a pinched first sacral nerve root (s1) cause pain when you sit down?

Possibly. A pinched s1 nerve root can send radiating pain down the leg to the outsife of the foot. Sitting does cause increased disc pressure which can increase the pressure on the nerve. A pinched nerve usually responds to convervative treatment-nsaid's, pnysical therapy and chiropractic treatment. Any loss of bowel or bladder control means heading to the er immediately. Consult your physician.
It may if due to a. Disc herniation as sitting places 4 times your body weight load across the discs of the lower back. Standing is about 2 times your body's weight and bending is the same as sitting. Lying down is the least load. Driving is worse than just sitting as cars and spines vibrate at the same harmonic frequency of 5 hertz so you are placing a vibrational stress on an already 4 times loaded disc seated.