Sure. You can try searching on major fanfiction platforms. Some writers like to explore unique relationships, and Percy dating Artemis could be one such creative exploration. Keep in mind that the quality and nature of these fanfictions can vary greatly. Some might be more focused on adventure and how such a relationship would work within the world of the series, while others might be more about the romantic aspect.
One common theme could be the clash of their different natures. Percy is a demigod with his own set of powers and a more human - like personality, while Artemis is a goddess with her own rules and a more aloof nature. Another theme might be the challenges they face from the other gods and goddesses who might not approve of their relationship.
Yes, there are quite a few. You can start by looking on popular fanfic websites like Archive of Our Own. Just search for 'Percy Dates Artemis' and you'll find a variety of stories with different interpretations and plotlines.
Percy could be an ordinary young man with a kind heart and a sense of adventure. Maybe he's a peasant who dreams of a different life and ends up meeting the princess by chance.
Percy could also react with a sense of determination. Instead of being defeated by the injury, he sees it as a challenge. He starts to figure out what made Artemis hurt him in the first place. If it was a misunderstanding, he tries to clear it up. If it was something more malicious, he starts to plot his revenge. He uses his own wits and the powers he has to either heal himself or find a way to make Artemis see the error of her ways. He might also seek the help of his friends, like Annabeth or Grover, to come up with a plan to deal with the situation.
The Hunters of Artemis are very likely to show up. They are Artemis's loyal followers and would surely be involved in the search for their mistress. Annabeth could also be there. She's Percy's friend and has great wisdom, which could be useful in devising a plan to save Artemis. Grover might be another character. His ability to sense things could help in tracking Artemis or finding clues about her whereabouts.
Maybe Artemis is on a hunt when she stumbles upon Percy. She might be initially cautious, as she's a goddess and Percy is a demigod. She could question him about his presence in Texas.