Can open mouth breathing cause facial pain problems?

Yes. Any "unnatural" position, including leaving the mouth open for too long or too wide can lead to tmd and facial pain. I would see a dentist or ENT if it continues.
Yes. Yes. There is usually an anatomical reason for mouth breathing. See ENt doc to check for nasal obstruction. Sometimes mouth breathing related to occlusion. If so, see orthodontist. If facial pain a problem, see TMJ-orofacial pain expert.
Open mouth breathing. Yes, open mouth breathing is a suspected contributing factor to temporomandibular disorders(tmd). The open position can cause strain on the ligaments and tendons which can ache after a prolonged period of time in a stretched position.
Yes. Mouth-breathing can cause facial pain as your jaw is stressed and not in it's comfortable position. This could cause muscle or joint discomforts and triggers pain.