Orochimaru could still be an antagonist. He might see Naruto as a threat especially if Kakashi is training him intensively. His goal of obtaining power could lead him to target Naruto.
Well, the plot could revolve around Kakashi providing Naruto with a more stable family - like environment. He might cook for Naruto, help him with his studies at the ninja academy in a more hands - on way, and also be there to support Naruto emotionally when he has nightmares about being alone. This could lead to Naruto growing up more confident and perhaps even developing new skills faster as he has someone constantly guiding him.
Danzo might be an antagonist. He's always been power - hungry and might see Naruto as a threat, especially if Kakashi is guiding and protecting him. Danzo could try to undermine Kakashi's efforts to raise Naruto and might even send his root agents to spy on them or cause trouble.
Well, one possibility could be that Kakashi takes Naruto under his wing and starts training him more intensively. He might teach Naruto some of his own special jutsu and also help him deal with his inner demons related to being an orphan. Another aspect could be Kakashi helping Naruto with his schoolwork at the ninja academy, which in turn helps Naruto gain more respect from his peers.
It would be very close. Kakashi would be like a father or an older brother to them. Naruto and Sasuke would respect and rely on him. They would probably have a more harmonious relationship with each other too, because Kakashi would be there to mediate any conflicts.
Kakashi would become more nurturing. In the original, he was a teacher, but as an adoptive father, he'd have to be more affectionate. He might show his face more often to Naruto, as a sign of trust, and be more open about his emotions.
Kakashi is known for his laid - back yet highly skilled nature. His influence on Naruto could be in terms of teaching him to be more strategic in battles. Rin, being kind - hearted and having a strong sense of care, would give Naruto the emotional support he lacks. So Naruto might become more balanced in his approach to life and ninja - hood.
At first, it might be a bit rocky. Kakashi is used to his own ways, and fem Naruto may be a bit headstrong. But as time goes on, they start to understand each other. Kakashi sees her potential and determination, and fem Naruto starts to respect Kakashi's wisdom.
One common theme is the idea of family. Since Kakashi is adopting the female character, there's a focus on creating a new family unit in the context of the Naruto universe. Another theme is growth, as the female character usually starts out inexperienced and grows into a strong ninja under Kakashi's guidance.
The fanfiction would likely show a different side of their development. Since Kakashi is a very experienced ninja, he could expose Naruto and Sasuke to different types of missions earlier. This would accelerate their growth not only in terms of combat skills but also in terms of understanding the ninja world. And with Kakashi as their adoptive father - figure, they would have a stronger emotional support system, which could potentially prevent some of the tragedies that happened in the original story.