Can I use moist heating pad to manage sciatica?

Yes. The sciatic nerve can be irritated because of a herniated disc, arthritis that results in tightening of the spinal canal, falls, injuries, or other rare reasons (tumors, infections). Anti-inflammatory medications, alternating heat/cold therapy, physical therapy (especially with traction) can be helpful to relieve sciatic pain.
Sure. However it will help only the skin or a little deeper into the muscle. If that helps your pain, then great. If not, consider seeing a pain/spine specialist to get other treatment options offered to you.
Yes. Some people do feel temporary relief with moist heat, though not all. You should use caution not to burn yourself, and never fall asleep on the heating pad. There other non-operative treatments such as stretching and core strengthening that can help as well. Should this fail, surgery can be very successful in alleviating leg symptoms.