In a Giyuu x Shinobu fanfic, the setting and plot can vary widely. Some might be set during the main events of the Demon Slayer universe, with them facing demons together. Others could be more of an alternate universe story, where they meet under different circumstances, like in a modern - day high school AU. The fanfic could also delve into their backstories and how those experiences shape their relationship. For instance, Shinobu's past with her sister could play a role in how she opens up to Giyuu, and vice versa for Giyuu's own traumas.
I would recommend 'Whispers of the Heart' for Giyuu x Shinobu fans. This fanfic focuses on their unspoken understanding. It's full of small moments that build up their relationship, like when Shinobu leaves a small flower on Giyuu's doorstep as a sign of her care. Also, 'Shadows and Sunlight' is worth checking out. It has a great plot that involves them having to go on a secret mission together and how that journey changes their relationship from colleagues to something more intimate.
The world would be in shock. The media would be filled with reports of his death. Civilians who had always seen him as a symbol of safety would feel vulnerable. There would be mass mourning in the cities he had protected.
Ryuk would probably just be amused. He is a Shinigami who has seen a lot in his existence. Fanfiction is just another strange human creation to him. He might chuckle at some of the more outlandish ideas in the fanfictions, like a character having superpowers beyond the Death Note.