No, it's not appropriate. Writing such sexually explicit fanfic involving underage characters (if they are in the original context where they are underage) is unethical and goes against moral and often community guidelines.
You can also write about their training. Maybe they are training together to improve their combat skills or their magical abilities. This way, you can still explore their characters in a positive and appropriate way.
One great Percy Jackson and Piper McLean fanfic is 'The Adventures Beyond Camp Half - Blood'. It shows their unique friendship and how they team up to face new threats outside the camp. It has a lot of action scenes and interesting character interactions.
A 'Percy Jackson shower fanfic' might be centered around Percy's internal thoughts and feelings as he showers. In the 'Percy Jackson' universe, he has a lot on his plate. So, in the shower, he could be reflecting on his latest adventure, his relationships with his friends like Annabeth and Grover. It could also explore how he deals with the stress of being a demigod in a private moment like this.
No. Writing fanfic that involves sexual content about fictional characters is inappropriate and violates the respect for the original work and its intended audience. It also goes against ethical and moral standards in most communities.
Most likely Percy Jackson himself would be a main character, as it's a 'Percy Jackson' fanfic. If it's following the typical pattern, Annabeth might also make an appearance, perhaps knocking on the door while Percy is in the shower and having a conversation through the door about their next mission.
The main characters are obviously Percy Jackson and Piper. Percy is known for his heroic deeds and his ability to control water. Piper has her own charm - speaking powers which are very important in the story.
Friendship is a very common theme. Since Percy and Piper are both demigods, their friendship is often explored as they face various challenges together.
You can start by looking on fanfiction.net. It has a large collection of 'Percy Jackson' fanfictions, including many that focus on Percy and Piper. Just use the search bar and type in relevant keywords like 'Percy and Piper' and you'll get a bunch of results.
One great 'Percy Jackson, Percy and Piper' fanfiction is 'The Adventures of Percy and Piper'. It really captures the essence of their characters. The author does an amazing job of creating new quests for them while still staying true to their personalities from the original series.