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.
The Arkham Knight's origin story is quite complex. He was created as a military-trained antagonist to Batman. He has a vendetta against Batman, partly due to his own past experiences and the events that led him to become this highly skilled and vengeful figure. He was trained in combat and strategy, making him a formidable foe for Batman in the Arkham series.
Yes, there are many Arkham Knight fanfics that are suitable for general audiences. You can find them on various fanfic websites by using filters or search terms like 'family - friendly Arkham Knight fanfic' or 'PG Arkham Knight fanfic'.
The Arkham Knight story is of a considerable length. It can take quite a few hours to complete, depending on your playstyle and how much you explore the game's side quests and extras.