The relationship turns into one of opposition in some fanfics. Harry, now independent, may be working towards goals that Dumbledore doesn't approve of. Dumbledore may see Harry as a loose cannon and try to control him from a distance, while Harry resents any such attempts. Their interactions become tense and full of unspoken accusations.
In the 'Harry Potter' fandom, this would be a very controversial concept. Dumbledore is often seen as a mentor and a powerful, positive figure. Having Harry Potter destroy him goes against the general perception of their relationship in the original series. However, in the realm of fanfic, some might explore such a dark and alternative take as a way to examine different character arcs and what - if scenarios, but it would likely be met with a lot of resistance from purist fans.
It could be that Harry finds out about Dumbledore's secrets regarding the Deathly Hallows. If Harry thinks Dumbledore was more interested in the Hallows for his own purposes rather than for the greater good, it might make him hate Dumbledore.
In some fanfics, Dumbledore having 'babies' is a very strange and often out - of - character concept. Usually, if this idea is explored, it might be in an alternate universe kind of setting. Some fanfics may use it as a shock factor or to create a really different kind of family dynamic within the wizarding world. But it's not a common or canon - like concept at all.
Hermione might support Harry if she also sees some flaws in Dumbledore's leadership. She is very logical and if Harry presents her with solid evidence, she could side with him. For instance, if Dumbledore's actions are putting students in danger in some way, Hermione would want to protect her fellow students.
No. The official 'Harry Potter' story by J.K. Rowling doesn't have any such plot where Dumbledore sells Harry fanfic. It's likely a fan - created or very non - canonical idea.
Well, it could be that Harry discovers some dark secrets about Dumbledore's past actions. Maybe Dumbledore made a deal with some dark forces long ago that Harry thinks endangers the wizarding world. So, in a fit of moral outrage, Harry decides to 'sack' him, which could mean removing him from his position of power and influence at Hogwarts.
Well, Harry might outsmart Dumbledore by using his unique understanding of the magical world in ways Dumbledore doesn't expect. For example, Harry has a more direct experience with some dark magic items due to his encounters with Voldemort. He could use this knowledge to make decisions that seem counter - intuitive to Dumbledore but are actually more effective in a particular situation.
He might give Harry detention. Harry could be made to clean the trophies in the trophy room or help Hagrid with some difficult tasks in the Forbidden Forest under strict supervision.
One possible reason could be that Harry breaks school rules in the fanfiction. For example, if he ventures into the Forbidden Forest without permission. Dumbledore is strict about the safety of students and such disobedience might lead to punishment.