Well, the story could develop in many ways. Maybe Naruto could be an orphan found by someone else in the village, like an old ninja who decides to train him in secret. This would change Naruto's backstory completely. He might not have the same kind of pressure or expectations that come with being the son of a famous Hokage. His character could grow in a different direction, perhaps being more independent from the start as he has to create his own identity without the shadow of his 'father' Minato.
Yes, there are likely some 'Naruto Son of Evil Minato' fanfictions out there. Fanfiction websites are full of various creative takes on the Naruto universe. You can start by checking popular fanfiction platforms like FanFiction.net or Archive of Our Own. Just search for the specific title or relevant keywords to find ones that match this concept.
Sakura could be a significant character. In the context of Naruto being Minato's son, her relationship with Naruto might change. She could be more in awe of him at first or have different expectations. Also, Sasuke is likely to be a main character. His rivalry with Naruto could take on new dimensions as Naruto has the added legacy of being Minato's son. Their battles and interactions might be more intense and influenced by Naruto's heritage.
The relationship can be developed by having Minato appear in Naruto's memories more vividly. For example, Naruto could start having flashbacks of Minato's training sessions or family moments that he didn't know about before. This would make Naruto feel closer to his father.
One possible fanfiction could be about Kagome somehow traveling to the Naruto universe. Maybe she meets Minato and they form an alliance. And she discovers that Naruto is Minato's son. Shippo could be a unique addition, perhaps he has some special abilities that are useful in the Naruto world.
One way could be that Naruto is an orphan found by someone else entirely. Maybe he's taken in by a group of traveling merchants who raise him. His skills and abilities could be developed through different means, like learning from the various people he meets during his travels with the merchants.
Well, in such a fanfic, Naruto could be the son of someone like Jiraiya. Jiraiya was very close to Naruto and taught him a great deal. He might have had a secret relationship that led to Naruto's birth. Another possibility could be that Naruto is the son of an unknown shinobi from a far - off village who somehow ended up in Konoha.
He could inherit Naruto's chakra - rich nature. So, he might have a large reserve of chakra like his father. Also, since Naruto has the Nine - Tails' chakra, Minato might have some sort of unique connection to it as well, perhaps being able to access a small portion of its power in a different way than Naruto does. And of course, he could inherit some of Naruto's signature jutsu like the Rasengan, but with his own twist.
Naruto would likely be more confident from the start. Since he has the combined blood of Minato and Mikoto, he might not have the same insecurities as in the original story. He would probably be more aware of his potential and be more self - assured in his abilities.