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 common plot point is that Naruto's sickness becomes a threat to the village. Since he's often seen as a protector, his being incapacitated can make the village vulnerable. Another is the way his friends react. They usually band together to support him, like Sasuke might return to the village out of concern for Naruto's health. Also, the search for a cure can be a big part of the plot. It could involve seeking out rare herbs or consulting with ancient medical ninjas.
His teammates, Sasuke and Sakura, would surely be involved. Sasuke, although he may not show it as much, still cares about Naruto. He would keep an eye on Naruto's condition, and if there was any sign of danger, he would be ready to protect him. Sakura would be more hands - on with the medical care, checking his temperature, giving him medicine, and making sure he rests properly.
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.
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.
He might slow down the progress of the mission. His teammates would have to take extra care of him. They could either find a safe place to let him rest and recover or struggle to complete the mission while protecting the sick Naruto.
Sakura, being a medical ninja, would immediately start examining Naruto. She'd be very worried and use all her skills to try and diagnose the illness. Sasuke, on the other hand, might not show his concern openly but would be constantly on edge. He might secretly look for ways to help Naruto get better. Shikamaru would probably start thinking of strategies to keep the village safe while Naruto is sick.
One plot could be Naruto getting a mysterious illness that no one in the village can figure out. His friends, like Sakura and Sasuke, start to search for a cure. They might go on a journey to find a rare herb or seek the help of an old sage. Another could be that the illness is a result of a new enemy's jutsu. As he gets sicker, the village starts to worry about their future without their hero.