Maybe in the fanfiction, Leila and Lelouch meet in a neutral zone by accident. They then engage in a battle of wits. Lelouch tries to manipulate her with his Geass, but Leila's strong will resists it. This encounter makes them both respect each other's abilities. As a result, they form an uneasy alliance to take on a common enemy within the Britannian military that is causing more harm than good to both of their causes.
Another possibility is that the main antagonist could be an organization within the Britannian Empire that is secretly pulling the strings. They might see Lelouch and Leila as a threat to their hidden agendas. This organization could be using underhanded tactics like assassination attempts, spreading false rumors, and creating internal conflicts within Lelouch and Leila's alliance to try and bring them down.
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.