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.
Often, there's a theme of justice being redefined. When Batman kills the Joker, it questions the traditional forms of justice in Gotham. It might also explore the theme of power. Does Batman become more of a tyrant or does he find a new way to use his power for good? And there could be a theme of redemption for Batman, as he tries to justify his actions to himself and others.
Yes, it can be. In fanfiction, Batman killing the Joker represents a major departure from the traditional relationship. It challenges the idea of Batman's no - killing rule and delves into a darker side of the Dark Knight. It can explore themes of vengeance, moral breakdown, and the consequences of extreme actions.
The Joker's death at Batman's hands could also have an impact on the other villains. Some might see it as a warning and become more cautious, while others might see it as an opportunity to fill the power vacuum left by the Joker. Harley Quinn, for example, could either be driven to seek revenge on Batman or be forced to reevaluate her own life of crime without the Joker's influence.
A fanfic where Batman kills Joker could be very complex. For instance, the Joker may have planned his own death in a way that would make Batman cross the line, in order to prove that everyone can be corrupted. Batman's killing of Joker could then send shockwaves through the criminal underworld and among the other heroes. Some may see it as a necessary evil, while others may view Batman as having become a murderer, leading to a great divide in the DC universe of the fanfic.
Occasionally, in alternate storylines or specific comic series, Batman has been depicted killing villains. However, this goes against his traditional code of not taking a life in the main continuity.
This is a rather complex question. Batman is clearly intelligent. He has been trained in multiple disciplines, from martial arts to forensics. He uses his intellect to solve crimes and protect Gotham. The Joker, though, is a different kind of intelligent. He has an almost insane creativity when it comes to devising his evil plans. He can read people very well and use their weaknesses against them. He doesn't follow the normal rules of society or logic, which in some ways gives him an edge. For example, Batman often has to rely on his moral code and the law, while the Joker has no such constraints. So, it's not really possible to simply say one is more intelligent than the other.