Yes, there are likely some out there in the vast world of fanfiction. Fanfiction writers often come up with unique and unexpected crossovers like this. However, the quality and popularity can vary greatly.
I'm not entirely sure which specific 'Joker graphic novel' you mean. But generally, the Joker's kills are often brutal and random. He might kill for fun, to prove a point, or just to create chaos.
It's a really wild idea. On one hand, Naruto has the power to take on the Joker. His chakra abilities like the Rasengan could potentially be a deadly weapon against the Joker. But on the other hand, the Joker is a master of psychological warfare, which Naruto may not be used to dealing with. In a fanfiction, it would be fascinating to see how the writer would make these two characters interact and how Naruto would ultimately defeat the Joker.
Yes, there are some out there. You can start by looking on popular fanfiction websites like Archive of Our Own or FanFiction.net. Just search for the specific tag'superman kills batman' and you'll find a variety of stories. However, the quality can vary greatly from one to another.
You need to consider their different backgrounds. Superman has his Kryptonian heritage and his experiences in the Injustice 2 world. The reader could be from a more ordinary background. Show how these differences can both create friction and also draw them together. For instance, the reader might teach Superman about the simple joys of Earth life that he's forgotten in his pursuit of justice, which deepens their connection.
One possible reason could be mind control. Maybe some villain has taken over Superman's mind and made him do it. Another reason might be a fundamental difference in their ideologies that has escalated to the point of no return.
Maybe a family member of a police officer killed by the Joker. This civilian has a strong sense of justice and a personal vendetta. They are not bound by the same rules as the police and are more likely to take extreme measures. They have seen the destruction the Joker has caused to their family and will stop at nothing to get revenge.
One common theme is chaos. With Batman gone, Gotham descends into utter chaos as the Joker runs amok. Another theme is revenge. Sometimes the story is about someone seeking revenge for Batman's death. Also, the exploration of the Joker's victory is a theme, showing how he relishes in finally killing his nemesis.
There are some out there. Some explore the dark and twisted nature of such a scenario, delving into the psychological impact on Gotham and the other characters. However, the quality can vary greatly. Some might focus more on shock value while others try to build a complex narrative around it.