In such fanfictions, Lucius might show his protection by shielding Draco from the Dark Lord's direct orders. For example, he could try to find ways to delay or modify tasks assigned to Draco that are too dangerous. He may also use his influence within the Death Eater ranks to keep Draco out of harm's way.
A common plotline is that Lucius protects Draco from the consequences of his mistakes. For instance, if Draco fails a mission for Voldemort, Lucius steps in to take the blame or find a way to fix the situation. Another plotline could be Lucius protecting Draco from the bullying of other students at Hogwarts. He might use his influence to make sure Draco has a relatively peaceful school life.
Harry Potter can be an interesting character to pair with them. It creates a lot of conflict and potential for character development. Given their opposing stances in the original story, it can lead to some really intense and thought - provoking scenarios in fanfiction.
In fanfiction, it might be due to a redemption arc for Lucius. He realizes the error of his ways in the wizarding war and starts to view Hermione, a muggle - born with great talent, as someone worthy of protection. He could also be protecting her from other dark wizards who might want to harm her because of her intelligence and influence among the light - sided wizards. This protection could also be a form of self - atonement for his past misdeeds against the muggle - borns.
Snape might be protective of Draco in fanfiction because in the original series, Snape has a certain loyalty to the Slytherin house. Draco is a young Slytherin, and Snape could see a bit of his own past self in Draco. Also, Snape was tasked with some difficult and dangerous missions by Voldemort. He might want to protect Draco from getting too deeply involved in Voldemort's dark plans to spare the boy from a similar fate as his own.
One reason could be the complex relationship they have in the original series. Fanfic authors often expand on that. Maybe Draco realizes Harry's importance in the fight against the real evil, like Voldemort, and so he becomes protective. It also adds an interesting dynamic to the story, making it more appealing to readers who love exploring different character relationships.
The theme of unexpected alliances is also common. Lucius and Hermione are not characters one would expect to be on the same side. But in these fanfictions, his protectiveness creates an alliance between them. This alliance might be formed due to a common enemy, or a shared goal that they discover along the way. It makes for an exciting read as they navigate their relationship while dealing with the outside forces.
They could start by having an honest conversation. Lucius needs to admit his wrongdoings and Draco needs to express his pain. It's a first step towards redemption.
In fanfiction, Lucius beating Draco could be a way to show a complex family dynamic. Maybe Lucius is portrayed as a strict and sometimes cruel father figure. It could also be a plot device to create drama and show Draco's struggle within his family environment.
Yes, there might be. Fanfiction platforms like Archive of Our Own (AO3) or FanFiction.net could potentially have fanfictions with this theme. You can search using those keywords on these platforms to find out.