Is this sign used for brushing teeth or brushing off dirt?
No, this specific sign is only for brushing hair. Brushing teeth uses an index finger moving back and forth in front of your mouth, and brushing off dirt uses flat hands brushing against your clothes or body.
Can this sign mean the noun 'hairbrush' as well as the verb 'to brush'?
Yes! In ASL, noun-verb pairs are very common. Usually, the noun 'hairbrush' uses a smaller, quicker double movement, while the verb 'to brush' uses a larger, more continuous sweeping motion.
Does it matter which side of my head I sign this on?
You should generally use your dominant hand and sign on the corresponding side of your head. If you are right-handed, use your right hand and sign on the right side of your head.