Typically ear mites cause itchy ear with dark granular ear discharge. An ear infection causes itchy, swollen, painful and smelly ears with discharge/pus; if an external ear infection descends into the middle ear then this can affect balance. hearing and pupil size. With the symptoms you are describing, I am more suspicious of a brain disorder, but an ear infection can be ruled out by otoscopic examination and microscopic analysis of any ear discharge.
Head shaking can indicate neurologic disease such as brain inflammation or infection. Valley Fever is a fungal infection common in AZ that can infect the brain. Bulldogs are also a breed that can develop a poorly understood disorder called head bobbing, but this is diagnosed by ruling out other more serious diseases. I recommend getting the behavior on video to help the veterinarian with further diagnosis and therapy; additionally a referral to the veterinary neurologist (www.vetneuro.com) is in order if bloodwork and Valley Fever testing are all normal.
Good luck,
Kimberly Coyner, DVM DACVD
www.dermvetvegas.com