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.
One common theme is the exploration of their relationship. Given their different personalities in the original work, fanfictions often focus on how they bridge the gap between them, like how Giyuu's stoicism and Shinobu's more cheerful yet calculating nature interact.
He probably went through a range of emotions. At first, he might have been in shock, unable to believe that such a thing had happened. Then, he would likely turn to his family and friends for support while trying every possible way to find his love. He could have also offered a reward for any information leading to the safe return of his love. He would be constantly worried about the safety and well - being of his loved one.
You can start by looking on fanfiction websites like Archive of Our Own (AO3). It has a large collection of fanfictions, and you can easily search for 'female giyuu fanfiction' using the search bar. Another option is FanFiction.net, which also has a good number of such stories.