Not exactly. Batman Begins is one interpretation of Batman's origin story, but it's not the definitive or only 'real' one. Different Batman stories and adaptations offer various takes.
Yes, Batman Begins can be considered a mystery fiction to some extent. It has elements of mystery as Bruce Wayne uncovers secrets and faces hidden threats in Gotham City.
One idea could be that the OC is a childhood friend of Bruce's who has a unique skill, like being an expert in ancient martial arts from a far - off land. They could influence Bruce's training and view on justice in a different way. For example, they might introduce him to more spiritual aspects of combat and self - discovery.