On Naruto's side, he would probably feel guilty after realizing what he had done. This guilt could make him try to make amends, but Sakura's initial reaction of anger and hurt would still be a big obstacle in their relationship. Their interactions would be filled with tension and awkwardness.
Kushina could be an important character even though she's dead in the original story. In this fanfic, her spirit or memories could play a role in Naruto's upbringing. For example, Minato could tell Naruto stories about Kushina's feisty personality and her love for the village.
In some fanfictions, Minato saves Naruto physically. For example, he might use his teleportation jutsu to appear right in the middle of a fight and whisk Naruto away from danger. In others, he could save Naruto spiritually. He might enter Naruto's mind, like in the fanfic 'Minato's Mind - Saving Mission', and give him the strength and confidence to overcome his inner demons. Sometimes, Minato saves Naruto from a political situation. For instance, in 'Minato and the Hidden Leaf's Politics', he uses his influence and wisdom to protect Naruto from being used as a political pawn.
Minato would likely be extremely shocked. Naruto is his son, and the idea of Naruto attacking him would be completely unexpected. He would probably try to talk to Naruto first, to find out what is wrong and why he is being attacked.
In fanfiction, Naruto and Minato often represent a connection between generations. Minato was a great ninja in his time, and Naruto is following in his footsteps. Fanfictions can explore how Naruto lives up to his father's reputation.
I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide variety of themes and scenarios, some of which might be rather strange or controversial. However, Naruto fanfiction often explores different relationships and alternate storylines, but this particular one seems a bit unorthodox and I haven't personally come across it.
Well, it depends on the specific fanfic. Some fanfics might really emphasize Naruto's growth to be on par with Minato in terms of smarts. They could show Naruto learning from Minato's memories or having some sort of spiritual connection that transfers knowledge. However, other fanfics might focus more on other aspects of Naruto's character and not necessarily make him as smart as Minato.
One possible reason could be for Naruto's protection. Maybe Minato knew of some great danger that was coming for Naruto and he thought that banishing him would keep him out of harm's way. Another reason might be related to some prophecy or hidden power within Naruto that Minato feared could be misused if Naruto stayed in the village.
Sakura could be another main character. She might be shocked by Naruto's arrest and start her own investigation. Maybe she teams up with Sasuke, who also has his own complex feelings towards Naruto. Their combined efforts could play an important role in uncovering the truth.
One popular theme is Minato getting to know Naruto in the afterlife. Since Minato died when Naruto was a baby, fanfictions often explore their relationship in a spiritual realm.