What can cause lower abdominal pain, swelling and sciatica pain?

Any number of causes: Within the abdomen or pelvis that may be compressing or irritating the lumbar nerves or plexus of nerves as they travel to form the sciatic nerve such as a diverticulitis, ovarian cyst as well as several other possibilities so you should seek an evaluation by your physician.

Pinched Nerve: Sounds like you have a pinched nerve in your back. I would suggest seeing a pain/spine specialist to evaluate you further to evaluate what level and what can be done to treat you. There are interventional treatments besides medications and surgeries that might reduce or eliminate the pain altogether.