It depends on the specific comic series or storyline. Sometimes, it's a complex situation where Batman's actions might have indirectly contributed to Dent's demise. But in others, it might not be as straightforward.
One possible scenario in Batman saves Batgirl fanfiction could be that Batgirl is in a dangerous situation, like being held captive by a powerful villain. Batman, with his detective skills and combat abilities, manages to find her location and battles the villain to rescue her. It's a classic superhero - damsel in distress (although Batgirl is also a superhero in her own right) situation that often plays out in fanfiction.
One of my favorite scenarios in Batman saves Catwoman fanfiction is when it takes place during a city - wide disaster. Batman has to fight through chaos to find Catwoman, who might be trapped in a collapsing building. It shows his determination to save her no matter what.
There are also fanfictions that explore the aftermath of the rescue. How does Catwoman react? Does she become more indebted to Batman or does it create some sort of power shift in their relationship? These stories can really dig deep into the complex relationship between the two characters.
Yes, there are quite a few. Some explore a different side of the Joker where he has a moment of redemption and saves Batman. It often plays with the complex relationship between them and the idea that even the 'villain' can do a heroic act.
One great 'Batman Saves Robin' fanfiction is 'The Dark Knight's Rescue'. It really delves into Batman's determination and the emotional bond between the two. The author does an amazing job of showing Batman's skills as he battles through various villains to reach Robin. Another good one is 'Batman's Swift Save'. It's a shorter piece but has intense action scenes where Batman uses his gadgets to save Robin from a very dangerous situation.
Yes, it is possible in fanfiction. Fanfiction writers often explore 'what - if' scenarios. They could imagine a scenario where Batman has some sort of precognitive ability that he didn't know about before. With this ability, he could foresee the attack on his parents and stop it. This would create a whole new narrative for Batman, where his origin story is completely rewritten and his relationships with other characters in the Batman universe would also be different.
One common theme is redemption. Jason often has a troubled past, and Batman saving him can be seen as a step towards his redemption. It shows that Batman still believes in him and is willing to risk everything to bring him back from the brink.
Lex Luthor is often a main antagonist. He might be the one who sets up the situation where Superman needs to be saved, like creating a powerful weapon or a devious plan to weaken Superman.
A theme that frequently appears is redemption. Sometimes Batman might be in a situation because of a mistake he made, and Superman saving him is a form of redemption. Also, the theme of heroism is prominent. Superman's act of saving Batman reaffirms their status as heroes and shows their selfless nature towards each other.