Obviously, Paul and Percy are going to be main characters. Paul, as the one who finds out Percy's big secret. Percy, being the demigod whose secret is revealed. And there could also be other characters from the Percy Jackson series like Annabeth. She might be involved in helping Percy deal with Paul's discovery, either by giving advice or by being there to support Percy emotionally.
In some fanfictions, Paul could end up getting involved in the demigod world himself. Maybe he discovers he has some latent powers or a connection to the gods. This would completely change his perspective on life and his relationship with Percy. He could become a mentor figure for Percy in the mortal aspects of dealing with his demigod status.
Paul could react with disbelief. He might think it's all some sort of elaborate joke or Percy's wild imagination. After all, the idea of demigods and Greek gods being real is quite far - fetched for a normal mortal like him.
Well, if Paul finds out in the fanfiction, he might be shocked at first. He could react in different ways depending on what he finds out exactly. Maybe he discovers Percy's true identity as a demigod. He might try to protect Sally more, or he could be scared and want to distance himself from the whole situation.
Yes, there are many. You can find them on various fanfiction websites. Some might explore Percy Jackson in an imperial or demigod - like role in new and exciting adventures that the original series didn't cover.
In this fanfiction, Harry could be thrown into a whole new world of Greek mythology. He might have to go on quests, just like the traditional demigods. His friends at Hogwarts would be really confused when he starts showing these new demigod - like abilities. And he'd have to learn about his new heritage, which would be a huge shock considering he thought he was just a wizard.
In a 'Percy Meets Paul' fanfiction, Percy could be a demigod, like in the Percy Jackson series. He has powers related to the Greek gods. Paul, on the other hand, could be a normal human who is very intelligent and curious. Their meeting could be a chance encounter, like Percy accidentally stumbling into Paul's life while on a mission. Or Paul could be someone who has been unknowingly drawn into Percy's world of magic and mystery.
I don't think it's a very common or popular theme. The Percy Jackson fandom usually focuses more on positive relationships and adventures among the characters. Choking is a rather dark and extreme concept that doesn't really fit the general tone of the fandom most of the time.