The Penguin could be a great antagonist. He might see Batman's daughter as a new threat to his criminal activities in Gotham. He could try to set traps for her, using his devious mind and his army of thugs.
One idea could be that Batman's daughter discovers a secret society in Gotham that her father was unaware of. She has to use her own skills, a blend of what she learned from Batman and her own unique abilities, to expose the society.
The number of Batmans in the comics varies. There are classic, modern, and alternate reality versions, so it's not a straightforward count. But it's safe to say there are multiple distinct depictions of the Caped Crusader.
There are quite a few different versions of Batman in the comics. It's hard to give an exact number, but there are variations based on different storylines and alternate universes.
One common theme might be the struggle between the expectations of her father's position and her own desires. Since she is the vicar's daughter, there could be a lot of pressure on her in the story.
In batmans first story, the most important elements are the origin of Batman. Bruce Wayne's loss of his parents is central. It led to his determination to clean up Gotham. His early days of becoming Batman, like creating his costume and getting his first set of gadgets. His first battles against the local criminals in Gotham also set the tone for his future as the caped crusader. His moral code, which was formed during these early experiences, is also a key part of the first story.
Batman's first story introduced us to the Dark Knight. It showed his origin as Bruce Wayne, a young boy who witnessed his parents' murder. This tragic event led him to vow to fight crime in Gotham. He trained himself to be a master of combat and detective skills. In the first story, we also saw his first forays into fighting the criminals that plagued the city, starting his journey as the iconic Batman.