A possible fanfiction idea is that the dog is a reincarnation of an ancient being that has a connection to both Lucifer and Crowley. In the past, this being was involved in a great battle between good and evil. Now, as a dog, it has no memory of its past. Lucifer wants to use the dog to regain his former glory, while Crowley wants to keep the dog away from Lucifer to maintain the balance of power. The story could follow the dog's journey as it discovers its true identity and has to make a choice between the two powerful forces.
Well, the dog could be a messenger. For example, it could be sent by some higher power to warn Crowley about Lucifer's upcoming plans. It might have a special mark or ability that makes it stand out. Like it can run between different realms where normal creatures can't, and it uses this ability to keep tabs on Lucifer and report back to Crowley.
One could be a fanfic where Lucifer tries to manipulate Crowley into doing something shameful for his own gain. Maybe Lucifer convinces Crowley to betray a mutual acquaintance in a way that makes Crowley feel ashamed of himself later.
In some fanfictions, Lucifer might be punished by being trapped in a desolate dimension. This is often a way for the writers to show that even a powerful being like him can't escape justice. For Crowley, he could be punished by losing his demonic powers. It makes sense as his powers are a big part of what makes him a threat in the Supernatural universe.
In some fanfictions, Lucifer's punishment is shown as being bound to a mortal body and experiencing all the pain and frailty that comes with it. For Crowley, his punishment may be that he has to constantly watch his carefully built demonic empire crumble around him. It's a cruel punishment for a character who values power and control so much. The writers use these punishments to explore different aspects of their characters and how they would react in such extreme situations.
One idea could be a fanfiction where Lucifer and Michael are forced to team up against a new, greater threat. Maybe an ancient evil that even their father couldn't have foreseen. They start off hating each other but as they fight side by side, they begin to understand each other's perspectives a bit more.
A really popular one is 'Lucifer and Crowley: A New World'. In this fanfic, they find themselves in a completely new world, perhaps a parallel universe. They have to figure out how to survive and also how their relationship changes in this new and strange environment. It has a lot of action and adventure elements as they encounter new threats and opportunities.
A great one is the fanfiction that delves into the backstory of Lucifer and his relationship with his brothers before he fell. It gives a lot of context to their current situation and makes you understand their complex family dynamics on a deeper level.
Yes, there are some. Some fanfictions focus on how Crowley's loyalty to Lucifer is tested in situations that involve shame. For instance, when Lucifer asks Crowley to perform tasks that go against Crowley's new - found sense of self in the series.
The plot is crucial. It could involve Lucifer getting hurt in a fight against other supernatural beings, and then a character like Castiel or Sam coming to his aid. The way the author builds the tension during the hurt part and then the relief during the comfort part makes it engaging. For example, if Lucifer is injured and on the verge of losing his power, and then a spell or a kind gesture restores his hope. This kind of plot development keeps the readers hooked.
One of the popular pairings is Michael/Lucifer themselves. Their complex relationship, being brothers and eternal rivals, lends itself well to shipping in fanfiction. Writers like to explore the dynamic of their love - hate relationship in a romantic or at least very intense emotional context.