The other villages might react with a mix of shock and suspicion. They knew Naruto was a powerful force for good in the ninja world. His death could be seen as a sign that something was amiss in Konoha. Some villages might even fear that Konoha was becoming unstable and could pose a threat. Characters like Gaara, who had a special bond with Naruto, would be deeply saddened and might try to find out the real reason behind Naruto's death.
One possible plot could be a dark turn in the story. Maybe Kakashi is under some kind of mind - control. He could be forced to fight Naruto by a powerful enemy who has taken over his mind. Naruto, being unprepared for such an attack from his mentor, gets killed. This would then lead to a series of events like Sasuke going on a rampage to find out the truth and avenge Naruto, and the whole ninja world being in shock as one of their most promising heroes is gone.
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.
Naruto would probably be one of the first to try and rescue Kakashi. He has a strong bond with Kakashi and would not stand by while his teacher is in danger.
Naruto is often a common character. Since Naruto has a strong bond with Kakashi, he might be involved in Kakashi's time - traveling adventures. Sasuke could also be there. Their relationship with Kakashi in the original story makes it likely for them to show up. And of course, Guy would be a great addition. His friendly rivalry with Kakashi would add some humor and action to the time - travel scenarios.
There is no way to say for sure who the author of the 'kakashi book story' is. It could be a new author trying to make a name for themselves or an established author writing under a pseudonym. Maybe it's a story from a small press or self - published, in which case it would be even more difficult to find out who the author is without further research.
Naruto could be often involved. Since he's Kakashi's student, he might be part of the events that keep repeating in the time loop. Kakashi could be trying to protect Naruto in different ways each time the loop occurs.
Naruto might be the better singer in terms of energy and enthusiasm. His loud and boisterous nature could translate into a powerful and engaging singing voice. He'd probably belt out songs with great passion, getting the audience pumped up.
Sasuke could be a great addition. He could be a rival to Naruto even at a young age, and Kakashi would have to deal with their constant bickering. His brooding nature would add an interesting dynamic to the story.