One common theme is loyalty. Sayoko is very loyal to her master in the series, and fanfics often explore this loyalty further, like in situations where she has to make difficult choices to stay true to her loyalty. Another theme could be her relationship with other characters, such as how she interacts with the main characters in new and different scenarios.
Well, you can try looking on fanfiction.net. There are often a good number of 'Code Geass' fanfics there, and you might be able to find ones specifically related to Sayoko. Another option is Archive of Our Own (AO3), which has a large collection of fan works and a good search function to help you narrow down to 'Code Geass Sayoko' fanfics.
Yes, it is. As a 'Code Geass' fan, it gives you more content related to the series. You get to explore the story further and might discover new aspects of the characters and the world.
Code Geass did start as a manga. However, its anime version became even more widely known and beloved. The manga provided the foundation for the rich story and characters that were further developed in the anime.
Code Geass started as an anime. The manga was based on the anime's success and came after to expand the story for fans. Some elements might differ between the two, but the anime was the original form.