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, being a prodigy, would use his own knowledge of sealing techniques to help Naruto better understand the bond with the Nine - Tails. Kushina, on the other hand, would use her own experiences as a Jinchuriki. They would create a safe environment for Naruto to explore and learn about his power. They might also have him practice meditation to get in touch with the Nine - Tails' chakra in a calm and controlled manner.
Maybe through new jutsu or abilities. The fanfiction could give Naruto some unique power that allows him to communicate or negotiate with the Nine - Tails on a different level. He could use this power to show the Nine - Tails a different perspective, like the potential for harmony between them instead of constant struggle.
Yes, in some fanfictions, Naruto is often depicted as having a very special connection to the Nine - Tails that goes beyond the normal relationship in the original story. For example, he might be able to fully control the Nine - Tails' power in ways not shown in the canon, or they could have a mental bond that allows for unique forms of communication and cooperation.
Well, initially, Naruto might show kindness towards the Nine - Tails despite it being a powerful and somewhat feared entity. He could try to communicate with it on a deeper level. The Nine - Tails, in turn, might be intrigued by Naruto's tenacity. As time passes, they begin to trust each other more. Naruto may share his dreams and fears with the Nine - Tails, and the Nine - Tails might start to see Naruto as someone worthy of its friendship.
Another idea is that Naruto's strong will and determination somehow reach Minato in the past. Minato, inspired by this unseen force which is actually Naruto from the future, finds the strength within himself to overcome a seemingly impossible situation. This could be during the Nine - Tails attack on Konoha, where Naruto's spirit gives Minato the idea to seal the Nine - Tails in a different way that doesn't cost him his life.
One great 'Minato Saves Naruto' fanfiction is 'A Father's Promise'. In it, Minato manages to break free from the death seal's constraints just in time to save Naruto from a powerful enemy. Another one could be 'Minato's Return', where Minato is brought back to life through some powerful jutsu and he immediately focuses on protecting Naruto. And then there's 'Saving His Son', which shows a very emotional journey of Minato coming back to save Naruto from a life - threatening situation.
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.
Another unique exploration could be through a shared goal that is not related to the main Naruto plot. For example, they both want to protect a special place in the spirit world that has significance for the Nine - Tails. To do this, they have to work together in a way that they haven't before. Naruto has to use his human skills and the Nine - Tails has to use its tailed - beast powers in harmony. This would show a different side of their relationship, one based on mutual respect for each other's abilities.
To develop the baby nine - tails, it could be given a unique personality trait such as being very empathetic. So, it can not only feel Naruto's emotions but also those of other villagers. This causes it to try and bridge the gap between Naruto and the villagers who fear it. It might go out of its way to show the villagers that it means no harm, which in turn would make for an interesting character arc as it tries to change the perception of the nine - tails in the village.