In some fanfics, Harry might hate Dumbledore because he feels Dumbledore has been using him all along as a tool to defeat Voldemort. Dumbledore has been making decisions for Harry without fully considering Harry's feelings and desires. For example, he didn't tell Harry about the prophecy until it was almost too late, and Harry might see this as a form of manipulation.
Maybe in the fanfic, Harry has his own ideas and beliefs that conflict with Dumbledore's. He might think Dumbledore's plans are too risky or not in line with his own sense of justice.
In some fanfictions, Dumbledore might bow out of respect for a powerful magical entity or a person with extraordinary abilities. It could also be a sign of his humility in the face of a situation where he realizes he is not the most knowledgeable or powerful.
Perhaps Harry was feeling betrayed or misunderstood and wanted to express his strong emotions directly to Dumbledore through a howler. This could be a way to make his point and demand a response.
It could also be for the sake of creating a different plotline. Writers might want to explore how Harry would develop without the support of his father figure. So they make James ignore Harry to see how Harry overcomes the challenges on his own.
Well, in the wizarding world, he was focused on more important things like fighting dark wizards, protecting his friends and family, and dealing with the politics of the magical community. Fanfic was just something that happened outside of his main concerns. It was like a side - show that he didn't have the time or inclination to pay attention to. He had enough on his plate without getting involved in the fictional creations of his fans.
One possible reason could be that fanfiction often takes the original story in directions that might not align with the author's vision. J.K. Rowling, who created Harry Potter, had a very specific world and story in mind. Fanfiction can sometimes distort or change elements of that world in ways she might not have intended. Also, from a legal and copyright perspective, it's a bit of a gray area, and she may have wanted to distance herself from it to avoid any potential legal issues or confusion about what is 'official' and what isn't.
Maybe he's too busy with his own adventures in the original story world. He has quests to go on, friends to save, and monsters to fight. Fanfiction might seem like a distraction from his real 'job' in the Percy Jackson universe.