I don't have the specific date for when the Egghead Arc started in manga off the top of my head. But you can probably find it by researching on manga-related blogs or by asking fellow manga enthusiasts.
Not exactly. Batman Begins is one interpretation of Batman's origin story, but it's not the definitive or only 'real' one. Different Batman stories and adaptations offer various takes.
It might be significant as it could show the rebirth of Batman, perhaps with a new take on his character. It may also explore the relationship between Batman and Robin in this new phase.
One possible aspect of a Batman and Harley fanfic in the Arkham universe could be their complex relationship. Batman is always trying to uphold justice, while Harley is often caught between her love for the Joker and her own sense of right and wrong. In a fanfic, the author might explore how Batman could potentially save Harley from her self - destructive path. For example, Batman could use his detective skills to show Harley the truth about the Joker's abuse towards her.
In the traditional Batman comics, the relationship between Batman and the Joker is not a love story. It's more of a complex adversarial relationship. The Joker is a villain who constantly tries to cause chaos and harm, and Batman is dedicated to stopping him.