My eyes get really watery during them. Haven't checked on flushing of the face/cheeks. I don't think my eye gets read. Haven't noticed a runny or stuffy nose. My left eye (where the pain is near) has been more closed/droopy than my right eye for a few years -- so not sure if that's related.
I would see a neurologist, you sound like you may be suffering from cluster headaches. They're intense, sudden sharp pain much like what you describe.
Do you have other symptoms such as a runny or stuffy nose, a watery eyes and/or swelling of both or one eye, a little drooping of your eye, flushing of the face/cheeks or redness of your eye during these attacks? Those also are symptoms of cluster headaches.