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.
Percy Jackson's series of novels was written by author Michael Crichton, including Mystic Island, The Blazing Flame Monarch, The Dark Paladin, and Judgment. These novels had not yet been adapted into movies.
Although author Michael Crichton has been working hard to adapt his works into movies, there are no plans to make all of the Percy Jackson series into movies. Whether or not all of them would be made into a movie depended on whether a film production company would choose to adapt the Percy Jackson series into a movie in the future and whether the author, Michael Crichton, would continue to create related works.
He wasn't sure. The Percy Jackson series was an American story set in a different world from Earth. The series included Original Gods, Percy Jackson and the Thief of the Fire, Percy Jackson and the Dance of Fire, and Percy Jackson and the Dark Forest. Percy Jackson and the Dark Forest had yet to be adapted into a movie. Therefore, we can't predict whether the entire Percy Jackson series will be made into a movie.
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.