He would probably be a bit of an outcast at first. Since his parentage is so different, other demigods might be wary of him. But once they see his skills and powers, they would start to respect him. He might teach them some unique combat techniques he learned from his Titan mother Rhea, like different ways of using his powers in battle.
A great plot idea could involve Percy finding out that there are other demigods like him, children of Rhea and an Olympian god. They form a secret group and together they must face a powerful enemy that is specifically targeting them because of their unique parentage. This could lead to Percy having to make difficult choices between loyalty to his new group, his Titan mother, and his Olympian father, as their interests may not always align.
There's also a fanfic called 'Poseidon's Promise'. In this one, Percy has to deal with a mysterious prophecy related to his father. The story is full of twists and turns. It delves deeper into the relationship between Percy and Poseidon, which is really interesting to read about. It also has some great action scenes with Percy using his water - based powers.
One idea could be Percy feeling betrayed by Poseidon and then finding a new family among other outcasts in the demigod world. He could form a group with them and go on adventures that are not related to Poseidon at all.
In fanfiction, their relationship could be developed by having Poseidon be more involved in Percy's daily life. Instead of just popping in during big events, he could be there for the small things too. For instance, helping Percy with his schoolwork that involves marine biology. They could also have a project together like building a new safe haven for sea nymphs. Through this, they would bond over creating something positive. Also, Percy could start to take on some of Poseidon's responsibilities in the ocean, with Poseidon guiding him. This would not only strengthen their bond but also show Percy growing into his role as Poseidon's son.
One possible plot could be Percy's discovery of his true parentage as the son of Kronos and Rhea. This could lead to a huge power struggle within the demigod world. Other gods and demigods might fear his potential power and try to either eliminate him or recruit him to their side.
The Titans who don't want Rhea to gain more power through Percy could be antagonists. Kronos, for example, might see Percy as a threat to his own plans for domination.
In some fanfictions, Percy being the son of Chaos and Poseidon is a really cool concept. It often gives him even more power. He might be able to control not just water like a son of Poseidon but also have some chaotic powers. For example, he could disrupt the order of things in a battle, or create portals to different realms which are abilities not typically associated with a normal demigod.
Yes, there are quite a few. You can start by looking on fanfiction websites like Archive of Our Own or FanFiction.net. Some might explore Percy's unique heritage as the son of both Gaea and Poseidon and how it affects his relationships with other characters in the Percy Jackson universe, like his friends at Camp Half - Blood or his enemies.
One idea could be Percy discovering his connection to Rhea in a dream. Then, he starts to have strange powers related to her that he has to figure out how to control.