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.
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.
Minato might show his protectiveness by always being near Naruto in spirit. For example, he could leave behind some kind of jutsu or power that activates when Naruto is in great danger, like a hidden seal that releases a powerful defense.
At first, Naruto and Kushina might start to distance themselves from Minato. They could avoid him and not talk to him as much. As time goes on, the hatred might grow deeper. Naruto could start to oppose Minato in public or in important village matters. Kushina might also try to influence others in the village against Minato.
There could be. Fanfiction often explores various 'what - if' scenarios, even the disturbing ones like Naruto being abused by Minato. But these are not in line with the positive image of Minato in the main story. Writers may create such stories to explore extreme character development or just for shock value.
Minato being alive in fanfiction could mean more guidance and protection for Naruto. It could also lead to new alliances and conflicts within the ninja villages.
Minato could be developed further by exploring his time as the Fourth Hokage more deeply. For example, show his political decisions and how they were influenced by the presence of the Kyuubi. Also, his relationship with other villages could be explored, especially in relation to the Kyuubi's power. Maybe he had to make some tough calls to protect the village from those who might want to use the Kyuubi for evil.
In fanfictions, Minato might cry for Naruto due to the missed opportunities. He missed Naruto's childhood, all those years of Naruto being alone. When Minato sees the kind of person Naruto has become despite all the hardships, it can be overwhelming and lead to him crying. He regrets not being there for his son.
One possible fanfiction could be about Minato and Kushina watching over Naruto from the afterlife as he grows into a great ninja. They could be his silent guardians, influencing his decisions in subtle ways.
The village of Konoha would also be affected. With Minato still alive, he could continue to lead the village and make different decisions. This might change the relationships between Konoha and other villages. For example, there could be more peaceful alliances as Minato is known for his diplomatic skills. Also, Kushina being alive would mean that the Uzumaki clan might have a stronger presence in the village, which could lead to new jutsu developments and a different power balance within Konoha.