In this fanfiction, perhaps Naruto is on a quest to learn more about Kisame. He could meet people who knew Kisame and through them, uncover secrets about his supposed father. This could lead to him having to make difficult decisions, such as whether to follow in Kisame's footsteps in some aspects or completely break away and forge his own path as a ninja.
Iruka could be an important character. He's always been like a father figure to Naruto, and now with this new revelation, he might have to help Naruto deal with it. Hinata could also be in the story, and her feelings for Naruto could be complicated by his connection to Kisame. And then there could be some original characters created just for this fanfiction, like a ninja who was a former ally of Kisame and has some information for Naruto.
There could be. Fanfiction is all about exploring different and often wild ideas. However, it's not a common or canon concept. In the original Naruto series, there is no indication of such a relationship. But in fanfiction, authors are free to play with the idea. They might show Naruto growing up with a different set of values or loyalties due to his relation to Kisame.
The plot could involve Naruto being shunned by the Hidden Leaf Village for some reason and Kisame, seeing an opportunity, takes him in. Kisame might see Naruto as a tool at first, but as time passes, he develops a sort of father - son bond with him. Naruto would then have to deal with the fact that he's now associated with the Akatsuki, which would put him at odds with his former friends. He could use this new position to try and change the Akatsuki from within, with Kisame's begrudging support.
There could be. Fanfiction is a vast realm where all sorts of ideas are explored. However, I'm not sure if there is a highly popular or mainstream 'naruto is kisame apprentice' fanfiction. It might be a more niche concept, but it's entirely possible that some creative writer has delved into this idea to create an interesting story about Naruto learning from Kisame, perhaps incorporating Kisame's unique fighting style and view of the ninja world into Naruto's growth.
A possible plotline is that Kisame left behind a child (Naruto) before his death. Naruto is then raised in Konoha, but he has strange dreams and instincts that he doesn't understand. As he gets stronger, he starts to show abilities that are not typical of Konoha - style ninja. When he finds out about Kisame being his father, he decides to explore the Akatsuki's side of the story. He could try to understand why his father was part of such a group. Along the way, he might meet characters who knew Kisame and learn more about his father's past, which in turn helps him understand his own place in the ninja world.
Well, it could be that Naruto's relationship with the other Konoha ninjas changes drastically. They might see him as an outcast because of his association with Kisame. On the other hand, Naruto might try to bridge the gap between Konoha and the Akatsuki due to his upbringing by Kisame. This would create a whole new set of conflicts and alliances within the Naruto universe.
Perhaps Naruto discovers that Kisame has a family he's been protecting in secret, and this new knowledge makes Naruto see Kisame in a different light. He then tries to help Kisame reunite with his family while also dealing with the Akatsuki.
Kisame would probably be very angry at first. Samehada is his weapon and has been with him for a long time. He might feel betrayed and try to get Samehada back by force.
One idea could be a story where Naruto and Kisame are forced to team up during a mission that has a greater threat to the ninja world. They start off distrusting each other but gradually come to respect one another's skills.
The plot could involve Naruto using Samehada to gain an understanding of the Akatsuki's motives. Since Samehada has a connection to the Akatsuki through Kisame, it might hold some secrets. Naruto could use it to track down other Akatsuki members or find out more about their plans.