One reason could be loyalty. Snape is a complex character who has his own loyalties. He might force Draco into situations to test his loyalty to Hogwarts or to the side that Snape believes is right. For example, if there's a threat from Death Eaters in the fanfiction, Snape could force Draco to choose a side, either by directly confronting the Death Eaters or by helping to protect the school.
Snape might force Draco to protect him. Snape has his own complex motives in the wizarding world. By forcing Draco to learn certain skills or follow certain paths, he could be ensuring Draco's survival. For example, in some fanfictions, Snape sees Draco as a key to maintaining some sort of balance in the Slytherin house or within the Death Eaters' influence.
Well, Snape might heal Draco because of his own sense of morality. Despite his dark exterior, Snape has a code of ethics. Draco is still a student, and Snape may believe that every student should be given the chance to survive and learn from their mistakes. Additionally, Snape could be acting on behalf of Dumbledore in some way. Dumbledore often had complex plans, and Snape was his agent. Healing Draco could be part of a bigger plan to keep the balance in the wizarding world.
A situation could be when Draco fails to follow Snape's instructions precisely in a potion - making task. Snape is a perfectionist when it comes to potions, and if Draco messes up the ingredients or the brewing process, Snape might yell at him out of disappointment. Another situation could be if Draco is caught bullying other students, especially those from non - pure - blood families. Snape, despite his own Slytherin loyalties, may not tolerate such behavior and yell at Draco to stop.
Perhaps Snape forces Draco to participate in dangerous tasks related to the wizarding war. He might believe Draco has a role to play in the fight against the Dark Lord. This could put Draco in a lot of danger, but also give him a chance to prove himself. Snape could use his position of authority to make Draco comply, like threatening to tell Draco's father if he doesn't. It's a complex situation that can vary a great deal depending on the fanfiction's plot.
Snape often uses his own experiences to comfort Draco. He's been through a lot himself, so he can relate. For example, if Draco is struggling with the expectations of his pure - blood family, Snape might share how he dealt with similar pressures growing up.
In fanfiction, Snape might punish Draco for getting involved with the wrong crowd. For example, if Draco starts to associate with Death Eaters - in - training or those with malicious intentions. Also, if Draco fails to complete his academic tasks or cheats in exams, Snape would surely take action. This is because Snape values knowledge and academic integrity in Hogwarts. Additionally, if Draco causes trouble during important school events or ceremonies, Snape would likely punish him to set an example for other students.
Snape could also punish Draco by deducting house points. Since Draco is in Slytherin, losing house points is a big blow to his pride and to the whole Slytherin house. This would make Draco the target of his housemates' ire, adding to the punishment's severity.
In the fanfiction, their relationship would be complex at the start. Snape, having his own demons and secrets, would be a bit of an enigma to Draco. Draco, on the other hand, would be dealing with the loss of his family's status. Snape would start by providing a stable home for Draco. As they interact more, Draco would notice Snape's small acts of kindness. Snape would encourage Draco to explore his own identity outside of the Malfoy name. This would lead to a closer relationship, with Snape teaching Draco about things like loyalty, sacrifice, and the true nature of magic, and Draco in turn bringing out a more human side of Snape.
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 great Snape and Draco fanfiction is 'The Unlikely Bond'. It delves into how Snape takes Draco under his wing in a way that's not shown much in the original series. It shows their complex relationship evolving over time, with Snape guiding Draco through some of the moral dilemmas he faces.