I'm not sure if there is such a specific fanfiction. Fanfiction can cover a wide range of very creative and sometimes out - of - the - ordinary ideas, but this one seems rather unusual. You might try searching on some large fanfiction websites to see if it exists.
The chances are probably quite low. This is a very specific and rather unorthodox concept that might not be very popular among fanfiction writers. Most fanfictions tend to focus on more common themes like character development, new adventures, or different relationships within the bounds of the original story's feel.
One consequence is a change in his physical strength. Kurama provided Naruto with a huge boost in power, so without it, he might not be as strong in terms of brute force. However, this could also force him to become more strategic in combat.
Well, Naruto might gain new and powerful jutsu. Kurama could teach him some unique chakra control techniques that no one else in the village knows. This would make Naruto even more formidable in battle.
Naruto would likely become more powerful. With the Yin Kurama, his combat abilities could increase significantly. He might also become more mature as he has to deal with the added responsibility of this new power. For example, he would need to be more cautious in using it so as not to harm others accidentally.
If Naruto gets rid of Kurama in fanfiction, one way it could change the story is that Naruto would lose a major source of his power. He'd have to rely more on his own natural skills and training. This might make him seem more vulnerable at first, but it could also lead to him developing new and unique abilities. For example, he could focus more on his Sage Mode training and become even more proficient in it without Kurama's chakra interfering or overshadowing his progress.
One common theme is justice. Naruto seeking revenge is often seen as a way to right the wrongs done to him or his loved ones. It's about making the guilty pay for their actions.
Well, Kurama might train Naruto by teaching him how to better control and access chakra. Since Kurama has a vast amount of chakra, it could show Naruto unique ways to harness and use it. For example, it could guide Naruto in creating more powerful and complex jutsu by using their combined chakra in a more efficient way.
There's also 'Revenge of the Jinchuriki: Naruto's Path'. Here, Naruto's motivation for revenge could be related to the discrimination he faced due to being a Jinchuriki. He decides to show the world that he's not a tool but a powerful force. The fic might explore his growth as he battles against those who have underestimated and mistreated him.