I feel your pain 

There are probably many other strange disorders we don't know. 
I had not heard about bells palsy before too. 
But i think what most people mean when talking about stroke is in some probably actually bells palsy.
Your definition about the face pain looks very much like the problem i have.
Even now with medication with every eye blink (really) it is like a small needle is stabbed in that part of my face.
But i don't care now because without medicine a normal life is really impossible. 
One of the reason the last months i wasn't here around was this condition. 
I didn't care anything, was depressed and frightened, 
just thinking-waiting in horror when the next pain attack will come.  
For example i had gallstone and surgery, a very complicated appendicitis surgery etc. 
But this damn disorder is the worst pain i had. I literally  screamed in the pain attacks. 
I never did this before.
Your teeth problem and bells palsy is related to the same nerve as Trigeminal neuralgia. 
Is the pain always present?
Hope you get a fast and easy recovery from that teeth problem Jordan.