Will stress cause bell's palsy?

Not typically. Bell's palsy is thought to be due to a viral infection of the facial nerve but the exact cause is often unknown. While stress can aggravate many neurological problems it is not thought to be a cause of bell's palsy.
No. Bell palsey is felt to represent an reaction to a prior viral illness where antibodies developed to fight off the virus attack the facial nerve on one side instead. This produces the one sided facial droop. Most recover in time with no specific rx. Stress may predate a stroke, but this palsy has no link to that condition.