For Naruto, you could show his growth in terms of his leadership skills when he's with Naruko and Kushina. He could learn to be more patient and understanding, especially when Naruko might have a different approach to things. For Naruko, her character could be developed by showing her struggle to be recognized in a world where Naruto already has a reputation. Kushina could be developed by exploring her motherly side more deeply, as she interacts with both Naruto and Naruko.
One possible reason could be a misunderstanding. Maybe Minato thought Kushina had done something to endanger Naruto when in fact she hadn't. In the heat of the moment, he lost his temper and slapped her.
The Naruto x Kushina romance in fanfiction can be quite interesting. Fanfic writers might focus on the idea of Kushina seeing Naruto grow up and be proud of him. They could have tender moments like Kushina helping Naruto with his studies or training. There could also be a plot where they face challenges together, like protecting the village from a new threat. This kind of fanfiction allows fans to imagine a different family dynamic for Naruto that he was deprived of in the original story.
Minato might be a common character. Since he is Kushina's husband in the original story, his presence can add an interesting dynamic. Also, Jiraiya could be there. He was Naruto's mentor and his interactions with the harem situation could be quite humorous.
In such a fanfiction, the relationship between Naruto and Kushina would be one of mother - son love and mutual respect. Kushina would be protective of Naruto and proud of the man he has become. Naruto would look up to Kushina and value her wisdom. With Hinata, Naruto's relationship would be that of a loving couple. Hinata would support Naruto in all his endeavors. When it comes to Kushina and Hinata, Kushina would likely see Hinata as a worthy partner for Naruto and Hinata would respect Kushina for being Naruto's mother and a strong kunoichi.
Their relationship could be developed through shared experiences. For instance, they could go on a mission together where they face a powerful enemy. During this mission, they would have to rely on each other's skills, which would deepen their bond.
In a fanfiction, if Minato were to slap Kushina, it would be a huge shock to the Naruto - Kushina relationship. It could potentially cause a rift between Naruto and his father Minato in Naruto's mind. Also, Kushina would likely be deeply hurt and angry, which might lead to her distancing herself from Minato. This event could also change the way Naruto views his family's past and his parents' relationship.
Another aspect could be that the fanfiction delves into how Kushina's presence affects Naruto's relationships with others. For instance, his relationship with Sasuke could be different. With a strong maternal figure in his life, Naruto might be more stable emotionally and be able to approach Sasuke in a more understanding way, which could potentially change the whole dynamic of their rivalry and friendship.
There's a fanfic called 'Naruto and Kushina: Unbreakable Bond'. This one focuses on the spiritual connection between them. It shows how Naruto, even without having known Kushina for long in the original story, still has an innate connection to her. The fanfic takes this connection and expands on it, creating a story full of family love and self - discovery for Naruto.
Yes, there are quite a few. You can start by looking on fanfiction websites like FanFiction.net or Archive of Our Own. Some of these fanfictions might explore a different kind of relationship between Kushina and Naruto, like what could have been if Kushina had survived. They often delve into the emotional bond that could have developed between mother and son in a more complex way, with added elements of love and care that go beyond the normal mother - son relationship shown in the original series.