Can facial pain can be effectively treated by a chiropractor?

No. Nothing can be effectively treated by a chiropractor. I understand that a lot of people believe otherwise - but unfortunately, there is no credible scientific evidence to support anything that chiropractors do. No flames, please; just telling it like it is.
Why? Why would you want to treat facial pain with chiropractic treatment? See a TMJ-orofacial pain expert for the best possible treatment. Any dentist can be a TMJ orofacial expert with the proper training and experience. Most commonly, oral surgeons, prosthodontists, and orofacial pain specialists. Ask your MD, your dentist and your dental society for referrals.
Not likely. Depending on the diagnosis, more than likely a chiropractor would not be able to help with facial pain.
No. Chiropractic theory postulates disease and pain originating in nerves from the spine. Facial pain is from nerves that enter the face directly through the skull. Chiro manipulating might make you relaxed but it will not treat your facial pain. You should start with a neurologist to make a proper diagnosis and start you on effective therapy.