In some fanfictions, it could be due to a special jutsu gone wrong. Maybe a jutsu that Kakashi was using to transfer some of his chakra to Naruto for a mission accidentally transferred his Sharingan power as well.
One interesting plotline could be Kakashi struggling to control the new power. Since the Rinnegan is extremely powerful, he might initially have a hard time adjusting. For instance, he accidentally activates its powers during a simple sparring session and causes a huge commotion. Another plotline could be how the other ninja react to Kakashi having the Rinnegan. Some might be jealous, some might be afraid, and some might see it as a new hope for the village.
Most likely not. Fanfics with such themes are often inappropriate and violate the general ethics and norms of the Naruto series. The Naruto series is mainly about friendship, growth, and ninja adventures, not such improper content.
In some fanfictions, when Kakashi finds Naruto, it could be a situation where Naruto is in danger. Kakashi would then rescue him and from that moment on, their bond becomes even stronger. Kakashi might then focus on helping Naruto control the Nine - Tails' chakra more effectively, which would change the course of Naruto's development in the story.
Another situation is when Kakashi is on a solo mission. He might encounter a group of powerful opponents who manage to wound him. This forces him to rely on his wits and basic ninja skills to survive while waiting for backup. It can show his resourcefulness even when injured.
Often, there are themes related to the setting or the world they live in. If it's a fantasy or science - fiction setting, their relationship might be affected by the unique elements of that world. For instance, if there are different species or powers, how they deal with the differences in their relationship. And sometimes, there could be a theme of sacrifice, where one of them has to give up something for the sake of the relationship.
Their relationship could develop very quickly. Kakashi might see Naruto as a prodigy waiting to be shaped. He would start mentoring Naruto intensively, and Naruto would look up to Kakashi like a father figure. They'd probably go on missions together more often, and Naruto would learn a great deal from Kakashi's battle strategies.
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.
Well, another option could be to check out some dedicated fanfiction forums. People often share their own 'gil gets a girlfriend' fanfictions there. You might find some really creative and interesting stories that way. Also, don't forget to read the reviews and ratings to ensure you get a good read.