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.
His personality would likely be more stoic. Kisame is a rather reserved and tough character, and growing up with him would probably make Naruto less exuberant. He might not be as quick to trust others as he is in the original story.
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.
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.
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.
His fighting style would be more stealth - based. Itachi was a master of genjutsu and Kisame was strong in close - combat and water - style jutsu. Naruto might learn to use genjutsu more effectively, which is something he wasn't very good at in the original story. He could also incorporate water - style techniques into his attacks, making his combat more diverse. Instead of relying mainly on his shadow clones and Rasengan, he might use illusions to disorient opponents before going in for a kill with a water - enhanced physical attack.
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.
Naruto's character development would be drastically different. Itachi's calm and strategic nature might temper Naruto's impulsiveness. He could learn to think before acting more often. Kisame's brute strength and loyalty could also shape Naruto. He might become more physically powerful in a different way than with Jiraiya's training. His view of friendship could change too. Instead of the large group of friends he had in the original story, he might be more reserved and have a closer, more exclusive bond with a few individuals, similar to the relationships within Akatsuki.
At first, it would be full of suspicion. Naruto is a Konoha ninja and Kisame is from the Akatsuki. But as the teaching progresses, they might develop a mutual respect. Naruto would respect Kisame's power and knowledge, and Kisame might respect Naruto's determination.
Well, in some fanfictions, Naruto might defeat Kisame in a battle and take the sword as a trophy. Maybe he uses his newfound Sage Mode powers or some other advanced jutsu to overpower Kisame and claim the sword.