Is stuttering caused by a brain injury?

Maybe. This speech rhythm and rate disorder affects 5% of children aged 2 - 5. It may last for several weeks to several years. Stuttering tends to run in families and genes that cause stuttering have been identified. There is evidence that stuttering may be a result of some unusual brain injuries, such as stroke or traumatic brain injuries.
One of causes. There are many causes of stuttering. You should consult a speech and language therapist for appropriate diagnosis and treatment.
Yes. Yes it could be. Stutering could be due to injury making a person unable to hear themselves.