Fanfiction often takes creative liberties. In this case, it might be that the author wanted to explore new power dynamics and character interactions. If Percy was Zeus' brother, it would completely change the power balance in the world of the series. Percy's connection to the sea and Zeus' control over the sky could lead to all sorts of interesting plotlines, like joint quests or power struggles between the two 'brothers'.
One possible plot could be that the entire Greek pantheon is thrown into chaos. Percy might be seen as a huge threat, and other gods and goddesses would either try to hunt him down or form alliances with him out of fear or a desire for power. Demigods would be forced to take sides, leading to massive battles among them.
Well, one possible plot could be Percy discovering he's Zeus' brother in a very unexpected way. Maybe through some ancient prophecy that was long forgotten. He'd then have to deal with the power struggles within the Olympian family, as well as his own identity crisis. Another aspect could be his relationships with other demigods changing, as being related to Zeus is a huge deal.
Another possibility is that Percy discovers some long - forgotten weakness of Zeus. In the world of fanfiction, the author could create a scenario where Percy stumbles upon an ancient prophecy or a hidden secret about Zeus. This knowledge could give Percy the upper hand. He could then use his wits and his powers to exploit this weakness and defeat Zeus.
In this kind of fanfiction, Percy might challenge Zeus over issues like the unfair treatment of demigods. For example, Zeus could be overly harsh on demigods who make minor mistakes while the gods themselves get away with much worse. Percy, being a hero, would stand up for his fellow demigods.
It's not possible in the traditional sense. Artemis is firmly committed to her status as a virgin goddess. But fanfiction often ignores such constraints. Maybe in a very creative fanfic, some sort of magical or divine intervention could make it seem possible, but it would be a huge departure from the source material. So while the idea is interesting for fanfic, it doesn't fit the normal Percy Jackson world.
One interesting fanfiction could be about Percy Jackson's adventures that are more closely tied to his relation as Zeus's grandson. Maybe he discovers some hidden powers that are passed down only through the Zeus lineage. He could face challenges where his Zeus - related powers are both a blessing and a curse, like when he has to control excessive lightning powers during a battle with a powerful monster.
It could also be about power. Percy might realize that with Zeus out of the way, he can reshape the world of the gods in a more just and fair way. He could believe that Zeus's rule is corrupt and that killing him is the only way to bring about real change.
His powers would likely become more god - like. He might gain the ability to control lightning, like Zeus. Also, his strength and speed could be enhanced to a much greater degree. His control over water, which was his main power before, could also be amplified and perhaps combined with his new lightning powers in some unique ways.
Well, at first, they might be cautious around each other. Percy is used to being the main hero, and the daughter of Zeus's sudden appearance could make him a bit jealous or at least wary. But as they face challenges together, they start to respect each other. Maybe she saves Percy from a difficult situation, and he realizes she's not just there to steal his thunder. Over time, they could become good friends, relying on each other's unique skills in battles against monsters.
For the demigods, things could change a lot too. Percy is a champion of the demigods, and his victory over Zeus might mean better treatment for demigods. They could be more respected and have more rights in the world of gods and mortals. However, it could also make them targets of those who are against Percy's actions.