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 great 'Batman Barbara Gordon Fanfiction' is 'Oracle Rising'. It really delves into Barbara's transformation into Oracle. It shows her strength and intelligence in a way that's both engaging and true to her character.
One more theme is the exploration of her dual identity. As Barbara Gordon, she is a librarian or a normal citizen, but as Batgirl/Oracle, she is a crime - fighter. Fanfictions play with this duality, showing how she balances her two lives. Sometimes she has to keep secrets from her friends and family, and this creates interesting plot twists and character dynamics.
Well, Gordon's reaction could be complex. On one hand, he might feel honored that Batman has chosen to disclose his identity to him. On the other hand, he could also be worried about the implications. Knowing Batman's identity means he has a greater responsibility to protect that secret. He might start to think about how this new knowledge will affect their working relationship and the safety of Gotham. For example, if the wrong people find out that he knows Batman's identity, it could put him and his family in danger.
Batman would approach the situation with his typical stoicism. He would be analyzing Spiderman's every move, trying to figure out if he can be trusted. Spiderman, on the other hand, would be like a breath of fresh air. He'd be bouncing around, full of energy and ideas. Gordon would be the voice of reason. He'd be the one to say things like 'We need a plan' or 'We can't just go in guns blazing'. Spiderman might come up with some wild ideas to take on the bad guys, Batman would refine them, and Gordon would ensure they could be carried out within the bounds of the law and with the support of the Gotham police.