One possibility is that they are in a mentor - mentee relationship. Gaara, with his experience and power, could be teaching Matsuri new techniques and ways to protect the village. Matsuri, being eager to learn, would be a great student for him. And this relationship could also evolve into a deeper friendship over time as they face challenges together in the fanfiction world.
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.
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.
A Naruto x Hinata fanfiction could focus on a mission they go on together during Naruto Shippuden. They are forced to rely on each other in a dangerous situation. Hinata gets injured protecting Naruto, and Naruto is filled with a new - found determination to protect her. This experience deepens their bond, and Naruto starts to see Hinata in a whole new light, leading to the start of their romantic relationship.
Matsuri's character could be developed as someone who is inspired by Gaara. She may strive to be like him in terms of strength and leadership. In fanfiction, she could be shown following Gaara's example in handling difficult situations. At the same time, her presence can also bring out a softer side of Gaara, as he might be more protective of her. Their relationship can be a tool to explore different aspects of both of their personalities.
One great Naruto x Hinata fanfic in Naruto Shippuden is 'A New Beginning'. It focuses on their relationship development after Naruto returns from his training. The story beautifully depicts how Hinata's feelings for Naruto become more obvious and how Naruto starts to see Hinata in a new light.
In many Naruto Shippuden fanfictions where Naruto gets sick, it often shows a different side of his character. Usually, his friends like Sakura and Kakashi would be very worried. Sakura might use her medical skills to take care of him. And his enemies might take this opportunity to try and attack. But Naruto's strong will would still be there even when he's sick. He might not be able to fight as powerfully, but his determination to protect the village would remain unshaken. This kind of fanfiction can explore the relationships in the Naruto universe more deeply when the main character is in a vulnerable state.
One great 'Naruto Shippuden Naruto and Sakura Fanfiction' is 'The Bonds We Share'. It delves deep into their relationship, showing how they grow closer during the Shippuden era. It has some really touching moments between Naruto and Sakura that stay true to their characters as seen in the anime.