Can a large cyst at left L4 L5 cause right lower back hip pain and sciatica?

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.
Yes it can. It can exert mass effect, and push the content of the spinal canal at the l4-5 level against the right sided bony structures, and hence cause right leg pain.
Depends. It depends where the cyst is. If it is in the facet joint or ligamentum flavum or along the nerve root, then it can cause hip pain and "sciatica" also known as radiculopathy.