There should not be an 'erotic' story involving Batman's mother Martha Wayne in the proper context of the Batman franchise. Batman stories are mainly about crime - fighting, justice, and the hero's journey. Any such inappropriate content goes against the essence and values of the Batman lore.
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 proper story is the origin story of Batman. Martha Wayne was Bruce Wayne's mother. They were a loving family until the tragic event of their murder in Crime Alley. This event was the catalyst for Bruce Wayne to become Batman, driven by a desire for justice. It's a story of loss and the birth of a hero.
Yes, in the Batman lore, there are many proper stories about Batman and his mother Martha Wayne. These stories usually focus on her death as a catalyst for Bruce Wayne's journey to become Batman, and how her memory and the values she instilled in him shape his actions as the Dark Knight.
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.