No. In the official Percy Jackson series, Percy is the son of Poseidon. Fanfiction often takes creative liberties and creates alternate storylines, but these are not part of the original canon.
Annabeth would be devastated. She and Percy have a deep connection. She might go into a state of shock at first, then be filled with a fierce determination to find out who was responsible and make them pay. She would probably also be racked with guilt, thinking she could have done something to prevent it.
There are many Percy Jackson fanfictions that incorporate a New York accent. Some might focus on how Percy's New York upbringing is reflected in his dialogue with that distinct accent. It could add a lot of authenticity and charm to the story. For example, when Percy is talking to his friends or getting into a scrape in the city, the accent can make his character come alive even more.
As of now, I'm not entirely sure. The Percy Jackson series has been very popular, and the author may well be working on a new novel. However, without any official word, it's hard to say for certain. It could be in the works, but it also might not be. Check fan forums and news sites related to young adult literature for any hints.
One possible fanfiction could be about Percy Jackson exploring the hidden magical corners of New York that are related to the concept of 'echo'. Maybe he discovers an ancient artifact in an alleyway that when touched, releases an echo of a past demigod's adventure. This could lead him on a new quest to figure out what the echo means and how it relates to the current threats against the demigod community in New York.
One idea could be that the new girl is a demigod from a very rare and powerful lineage. For example, she could be a descendant of a primordial god that no one has seen in centuries. This would bring a whole new dynamic to Camp Half - Blood as everyone tries to figure out her powers and where she fits in.
Imagine a fanfic where Percy has to navigate the New York City traffic while being chased by a pack of hellhounds. He has to use his water - based powers in creative ways, like creating water slides on the streets or using sewer water to slow down the hellhounds. This would be a really action - packed and fun story.