Does Suboxone cause permanent brain damage?

Not permanent. Brain damage is not the word I would use. Opiate addiction in general can cause long lasting changes to the brain. Suboxone is a type of opiate and completely stopping it can be difficult without other forms of support. It takes time, "months" for the brain to recover. Things such as stress or "triggers" even months to years down the road could cause cravings, possible relapse, meetings help!
No. And it doesn't cause ANY brain damage, so I am not sure where you got this question. Opioids do have effects on nerve cells, but those effects are ultimately reversible, and there is no evidence that use of opioids has any permanent effects of cognitive or motor function in human beings.
Brain Damage? People with opiate addictions have, in a way, already damaged their brain. High doses of opiates over a long period of time changes the brain. That is reflected in tolerance and dependency. A person will be able to tolerate higher and higher doses of opiates with chronic use, compared to when they first took opiates. Suboxone, if taken properly and for a long enough time, can reverse such changes.
Not sure but.... Depends how you define "brain damage". Present data suggests Suboxone is more "user friendly" in terms of causing changes in brain function and behavior than a pure opioid pain medication. No definitive study linking long term use of Suboxone with "brain damage".