Yes, there are some. You can try searching on fanfiction websites like FanFiction.net or Archive of Our Own. Some fanfictions might focus on their relationship before the tragedies in the Naruto story, exploring their growing affection and shared dreams.
In some Naruto fanfictions, there might be such scenarios depicted, but it's not part of the original Naruto story. Fanfiction often takes creative liberties and explores different relationships and interactions that may not be in the official canon.
Another theme is redemption. Maybe Yahiko was on a dark path or in a difficult situation, and Naruto's save gives him a chance to redeem himself and strive for a better future. It also reflects Naruto's own belief in giving people second chances.
Sure. The characters are a huge part of Naruto Shippuden fanfiction. Fanfic writers use the characters to attract readers who are fans of those characters. They might focus on a character's backstory that wasn't fully explored in the anime and expand on it. Also, the interactions between characters in the original show give fanfic writers ideas for different scenarios in their stories.
By their unique abilities. Naruto's Rasengan, Sasuke's Sharingan, these abilities can be the starting point for many fanfiction ideas. Writers can think about how these abilities could be used in different battles or situations not shown in the original.
One of the most popular is 'Naruto Shippuden: Sakura Hiden - Thoughts of Love, Riding Upon a Spring Breeze'. It's popular because it shows Sakura in a new light, with her own adventures and growth.
Well, in such a fanfiction, Naruto might grow up with different values. Yahiko's influence could make Naruto more radical in his pursuit of peace. He might be more willing to take extreme measures earlier on.
One common theme is 'lost love' since Yahiko dies in the original story. Fanfictions often explore how Konan copes with this loss and her lingering feelings for him. Another theme is 'what could have been', imagining a different future if Yahiko had survived.
His character might be more hardened. Growing up with Yahiko, he'd be exposed to the harsher side of the ninja world earlier. So he'd be less likely to trust blindly.
I don't think it's strange at all. Naruto is a very curious person. Fanfiction can offer new perspectives on his own life and the lives of his friends. He might learn something new about himself or get inspired by the creative ideas in the fanfiction. It could also be a form of entertainment for him during his downtime.