In fanfiction, Batman and Professor Pyg can be put in various scenarios. One could be a team - up of sorts, where Batman has to work with Professor Pyg (perhaps under some mind - control or blackmail situation) against an even greater threat. This would create a very tense and complex dynamic as Batman tries to both deal with the external threat and free himself from Professor Pyg's influence.
Professor Pyg's origin story is quite dark. He was originally Lazlo Valentin, a brilliant but deranged scientist. He became obsessed with the idea of creating a perfect society through extreme and inhumane means. He started experimenting on people, surgically altering them to fit his warped vision of an ideal being, which led to the creation of his 'Pyg-men'. His actions are a blend of his scientific knowledge gone horribly wrong and his twisted view of the world.
One common theme is the exploration of madness. Professor Pyg is a symbol of insanity, and fanfictions often show how Batman copes with such extreme madness. Another theme is the protection of Gotham's innocent. Batman is always trying to save the people from Professor Pyg's evil plans.
The key element is Lazlo Valentin's deranged mind. His over - inflated ego and belief that he could improve society through extreme means. Another is his scientific knowledge which he misused for his cruel experiments. Also, his lack of empathy for his victims as he turned them into Pyg - men without any regard for their well - being.
Probably. The Arkham series has a connected storyline, and it would make sense for the Arkham Knight new story to be a continuation. This way, it can keep the lore consistent and develop the characters further. For example, we might see how Batman has changed since the last encounter with his foes.