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.
Well, normal Naruto Shippuden fanfics are likely to be centered around the adventures, growth, and battles of the characters. They might explore different what - if scenarios within the framework of the Naruto universe. But 'naruto shippuden diaper fanfic' is likely to be more of a novelty or a very specific and strange take on the characters. It might not have the same depth or connection to the original story as normal fanfics do. For example, normal fanfics could focus on how a character overcomes a difficult battle, while a 'diaper fanfic' would be more about a really odd situation involving diapers which has little to do with the main Naruto themes.