A plot could involve Peter's identity being at risk and Daredevil helping him keep it a secret. Since Daredevil is so good at operating in the shadows, he could teach Peter some new ways to move around the city without being detected. They could also have a conflict about their different approaches to justice. Peter is more about saving everyone, while Daredevil has a more 'punish the guilty' attitude at times.
Sure, it can be normal. Just like any person might be interested in what others write about them or their alter - ego. It's a way to connect with the fan community on a different level.
Well, it could be a way for the daredevil to see how others perceive their character. Maybe they find it interesting to see the different scenarios and relationships that fans come up with in the fanfiction.
In many fanfictions where Hermione confronts Ginny Weasley, it often stems from a conflict over a boy, like Harry. Hermione might feel that Ginny is being too forward or not taking Harry's feelings into account. For example, if Ginny is making big moves on Harry without considering how it affects his studies or his mental state, Hermione, being the practical and studious one, could confront her. It could also be about their different personalities clashing in the context of the wizarding world's social dynamics.