Do you think that pain management can help the headache problem if infact it's caused by bad cervical stenosis (injections)?

Yes. Facet injection stenosis usually causes arm pain headaches caused by protective muscle activity treat upper facets in cervical spine.
Perhaps. Cervical stenosis can cause neck and arm pain. In more severe cases, it may compress the spinal cord enough to produce legs and bowel/bladder problems. Injections can be very helpful in reducing neck and arm pain, but it will not help with compression of the cord. Check with your spine specialist to see if you have myelopathy. Surgery will be required for that condition.
Yes. Cervicogenic headaches are often improved greatly with neck injections.