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.
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.
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.
One common theme is friendship. The strong bonds between Minato, Kakashi, Obito, and Rin are often explored in great detail. For example, how they support each other during tough missions.
One interesting Minato disciplines Naruto fanfiction could be where Minato teaches Naruto advanced sealing techniques. Minato was a master of sealing, and in the fanfiction, he could pass on this knowledge to Naruto in a strict yet loving way. This would not only strengthen Naruto's skills but also show a deeper bond between father and son.
Time - travel is also a frequent theme. For example, Minato might be brought to the present or Naruto sent back in time to meet his father. This allows for a lot of interesting interactions and character development. And of course, the theme of mentorship is there. Minato was a great shinobi, and fanfics often show him mentoring Naruto and passing on his skills and wisdom.