In some versions of the fanfic, Harry might have a sort of instinctive understanding that his parents had a good reason for leaving him there. So, while he's still not happy about it, he might be a bit more accepting. Maybe he can sense some sort of protection spell around the Dursleys' house that gives him a small sense of security.
Hermione might go with him. She's always been by Harry's side, and she would likely follow him out of loyalty and also because she might see the value in whatever mission or journey Harry is embarking on. She could provide the intellectual support and research skills that Harry needs.
Yes, there might be. Some fanfics like to subvert the expected negative situation. In these fanfics, even though the start is that Harry's parents hate him, through a series of events such as Harry showing extraordinary courage or kindness, his parents could have a change of heart. For example, he might save them from a great danger, which makes them realize their mistake and start to love him. This kind of plot twist can add more depth and surprise to the fanfic.
In these fanfics, a common theme is growth. Harry has to grow up faster and become more independent because of his parents' hatred. Another theme is the search for identity. Without the positive influence of his parents, he has to define himself. And the theme of resilience is also prominent. Harry has to be strong in the face of his parents' negative feelings and all the difficulties that come with it. He has to keep going and not let their hatred break him.
In some fanfics, they might abandon Harry because they are under a dark spell. This spell could make them lose their love for him and force them to leave. Another reason could be that they are blackmailed by some powerful dark forces. These forces may threaten their lives if they don't abandon Harry.
In certain fanfic scenarios, Harry's parents may have left behind some enchanted objects. These objects could have special powers. When Harry discovers them, they might activate and create a magical portal or something similar that allows him to leave the Dursleys' place. For instance, a locket that when opened, transports Harry to a safe place where his parents' magic has created a haven for him.
Another reason could be that they had some sort of premonition or prophecy that indicated the Dursleys' place was a crucial part of Harry's future protection. Maybe a seer told them that for Harry to grow into the wizard he needs to be, he has to endure the Dursley experience first.
One possible scenario could be that Harry leaves his parents due to a mysterious prophecy that he needs to fulfill away from them to keep them safe. Another could be that he discovers a hidden power within himself that he can't develop while staying with his parents, so he decides to leave.
Voldemort could still be the main antagonist. Even if Harry leaves his parents early, Voldemort might be hunting for him due to the prophecy. His dark presence would be a constant threat, driving Harry's actions and the plot of the fanfiction.
In some fanfictions, magic might be used to bring them back. For example, a powerful time - turning spell could be involved, allowing them to be retrieved from the past before their deaths. Another way could be through some sort of deal with dark forces, although this would likely have many negative consequences.