Yes. Either a neurologist or perhaps an ENT specialist as they treat many patients with facial palsy.
Bell's palsy. If it's bell's any pcp or er doc can treat this. If it's a stroke then you need a neurologist asap.
Yes, or... An otolaryngologist/ ent. Immediately.
ENT or neurologist. The sooner you are treated by either an ENT doctor or a neurologist with high-dose steroids (if indeed it is bell's palsy) the better you will do as far as recovery time and completeness. Ideally you should be treated within 3 days of symptoms.