Kakashi has a very sad story. He lost his father at a young age. His father was a great ninja but was shamed and ultimately took his own life. Then, Kakashi lost his comrades during a mission. His friend Obito 'died' right in front of him while giving him his Sharingan. These events have haunted Kakashi and made him a rather melancholic and complex character.
Kakashi had a love with Rin. However, Rin was captured by the enemy. Eventually, Rin sacrificed herself in front of Kakashi. This left Kakashi with deep sadness and a broken heart, forever haunted by the loss of his love.
Kakashi had a tough life. His father passed away early. In his ninja career, he witnessed the death of Obito and Rin. These events are the core of his sad story.
Well, first, Kakashi and Rin had a bond during their ninja training days. As they got involved in missions, danger lurked. Rin was kidnapped by the enemy. When Kakashi found her, the situation was already out of control. Rin chose to sacrifice herself rather than be a threat to Kakashi and their village. This series of events is what makes up the sad Kakashi love story.
Often, Gai is paired with Kakashi in Hurt Kakashi fanfiction. Gai, with his unwavering belief in friendship and his eternal rivalry with Kakashi, can be a great character to interact with a hurt Kakashi. He might try to pull Kakashi out of his slump through their friendly competition. Also, Rin is sometimes included. Since she has a significant past with Kakashi, her presence in the fanfiction can either be a source of comfort or more pain, depending on how the story is written.
In such fanfiction, the Kakashi OC often has unique physical features. For example, they might have distinct hair color or eye shape that sets them apart.
Kakashi might be initially surprised. But then, he would probably use it as a teaching moment for Naruto. He could say something like 'Naruto, stealing is not the way of a ninja.' And then give him a punishment like extra training.
Often, Naruto is paired with Kakashi in such fanfictions. Their bond from the original series can be explored in new ways in the context of Kakashi's reincarnation. For example, Naruto might be the one to discover Kakashi's reincarnated form and help him adjust.
Kakashi could use his Sharingan to analyze Obito's moves and find a weakness in his defenses. Since he has the Sharingan, he might be able to predict Obito's attacks and counter them effectively, eventually finding an opportunity to break free.
Kakashi's reaction would be complex. He is a man who has lost many people in his life. But Naruto was special. He would hold Naruto tightly, his hands trembling slightly. He might try to use some of his own chakra to keep Naruto alive a little longer, but deep down he knows it's futile. His face would show a mixture of pain, regret, and love. He would remember all the times he trained Naruto, all the lessons he taught him, and how Naruto had surpassed all expectations.