Batman is very brooding and methodical. Arrow, on the other hand, is more impulsive. During the rescue, Batman might be constantly trying to keep Arrow from doing something rash. For example, Arrow might want to immediately attack the villains when they are first spotted, but Batman would hold him back and wait for the perfect moment to strike.
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.
One popular 'Batman Arrow' fanfiction could be those that focus on a team - up between Batman and Green Arrow. They might face a common enemy like Ra's al Ghul who has plans that threaten both Gotham and Star City. In these stories, the different fighting styles and moral compasses of the two heroes are often explored.
I really like 'Green Arrow in Gotham'. In this fanfiction, Green Arrow finds himself in Gotham City and has to navigate the dark and dangerous world while also dealing with Batman's somewhat cold welcome. It's full of action and great character interactions.
A great Arrow and Batman fanfiction could involve the two of them training together. Batman sharing his detective skills with Oliver and Oliver teaching Batman some of his more acrobatic and archery skills. This could lead to a stronger bond between them and help them both in their respective cities' crime - fighting efforts.