Well, one main event is his relationship with his father. His father was shamed for choosing to save his comrades over a mission. This had a big impact on Kakashi. Another important part is his time with Obito and Rin. Obito's 'death' was a turning point for Kakashi, and he got his Sharingan from Obito then.
The key events in Kakashi's back story start with his father's misfortune. Sakumo Hatake's death left Kakashi with a sense of duty and the need to be a perfect ninja. Then, the mission where he, Obito and Rin were involved. Obito's sacrifice and Kakashi's gain of the Sharingan. His time in ANBU, which exposed him to the darker side of the ninja world. And later, his role as a mentor to the new generation of ninjas, like Naruto, Sasuke and Sakura.
The episodes that focus on Kakashi's back story are mainly in the arc where his past is explored. But it's a bit hard to pinpoint specific ones exactly.
Well, Kakashi's back story involves his early days as a ninja. His father's death had a huge impact on him. He was then trained hard to be a great ninja. He was put in a team with Obito Uchiha and Rin Nohara. During a mission, Obito 'died' (or so it was thought at first) and Kakashi got his Sharingan. This event also added to his complex character. He had to deal with the loss of his friend and the new power he got. He later became a mentor to Naruto and his friends, using his own experiences to guide them.
Another great Kakashi back in time fanfiction idea is him going back to the time right before Obito's 'death'. He could then change the events that led to Obito's fall to the dark side. This would rewrite the entire future for the Ninja world, as Obito played such a crucial role in many of the later events, including the Fourth Ninja War.
Kakashi's back story influenced his character in multiple ways. The death of Obito and Rin filled him with guilt and a sense of responsibility. He started to value his comrades more, which was a big change from his earlier more mission - focused self. His experiences also made him more empathetic towards others. He saw the pain and loss that war could bring, and this changed his approach to both life and combat. He became a more complex and nuanced character as a result of his back story.
With Minato, Kakashi's relationship could also shift. If Sakumo is alive, Kakashi may not be as eager to prove himself in the same way. Minato might see a different side of Kakashi, one that is more at peace with himself. This could lead to a more mentor - mentee relationship based on mutual respect and understanding rather than Kakashi's need for validation.
Since Naruto has more knowledge of the future, Kakashi might start to rely on him more, which could shift their relationship. Naruto could also see a different side of Kakashi from the past, and this new understanding could deepen their connection. For example, they might have to face some challenges together that make them trust each other more than before.
Well, one possibility is that they could use their future knowledge to prevent some of the tragedies that occurred in the original timeline. For example, they might be able to save some characters who died. They could also change the relationships between different ninja clans, making the ninja world more harmonious. And perhaps they would train harder in the past to become even more powerful ninjas in the future.