Well, there could be a fanfic where Lucifer forms an unexpected alliance with an exorcist. This alliance could be born out of a mutual understanding of a greater threat that is looming over both the demon and human worlds. They have to put aside their differences and work together, which makes for a really exciting story.
You can try fanfiction.net. It has a large collection of fanfics from various fandoms, including 'ao no exorcist', so there's a good chance of finding some great lucifer - related fanfics there.
Power is a major theme. Two Lucifers would bring a lot of power into one story, and it can be about who has more power or how they use their power together.
In the world of 'lucifer meets lucifer fanfiction', a common plot could involve a clash of egos. When two such powerful and complex characters meet, there's bound to be conflict. Some stories may also explore a romantic or at least a deeply emotional connection between them, as they are essentially the same being in different forms, and this connection can be both fascinating and disturbing at the same time.
He wasn't sure. The information I know about the source and background of the Ao Ao awaiting arrest comic is very limited. It could be adapted from a novel or an original. If you can provide more information about the manga, I will try my best to answer your questions.
Lucifer and Satan were not the same person. Although Lucifer was called the Morning Star, it did not explicitly indicate that he was Satan. Satan was the name of the ancient snake, and Lucifer was an independent existence. Therefore, Lucifer Morgenstern was not Satan.
I'm not entirely sure specifically what 'lucifer fiction' refers to. It could potentially be a genre of fictional works that somehow involve the concept of Lucifer, perhaps in a religious, mythological, or even a more creative and fictionalized context.