Can caffeine cause facial pain / stiffness emanating from the trigeminal nerve?

Unlikely. Unlikely. See oro-facial pain specialist for exam/diagnosis.
Yes. Caffeine can cause the symptoms you mentioned. Try to decrease the amount of caffeine to determine if this helps with your facial pain. If it does not, then you may need to go to the doctors to be evaluated and diagnosed properly.
Contributing factor. Facial pain is usually not caused by one thing but can have a number of things that can contribute. Certainly caffeine can excite the nervous system increasing the pain.
Probably. Caffeine is an interesting compound; it is known to trigger pain in the jaws and head as well as alleviate such pains. You can always try to reduce caffeine intake and see if it helps.