Can a brain injury on the left side cause pupil changes in the right eye?

Yes, but.... If you are having these issues, you should be in an icu. This is serious. Pupil changes from head injuries are likely a precursor to a fatal event. If this is chronic, i rec a full neurological exam, then an eye exam by an md.
Depends. It depends where on the left side the injury occurs, but theoretically yes...