Another reason might be that Harry is suspected of sneaking around in areas where he's not supposed to be. Snape, as a vigilant teacher, may punish Harry to maintain discipline in the school. He may think Harry is up to something related to the ongoing dangerous situations in the wizarding world and wants to keep him in line.
In some fanfictions, Snape might punish Harry by giving him extra detention tasks like scrubbing the cauldrons in the Potions classroom until they shine. This is a common form of punishment as it's within Snape's domain as the Potions master.
In some fanfics, Snape might punish Harry because of his perceived insolence in class. Harry has a rebellious streak, and Snape, being strict, wouldn't tolerate any behavior that he deems disrespectful to his authority as a teacher.
However, as the story progresses, Harry may start to understand the reasons behind Snape's actions. He could become more resigned and accept the punishment as a part of his growth. He might use the experience to become more cautious in his actions and more respectful of the rules.
At first, Harry is likely to be angry. He might feel that Snape is being unfair and targeting him. Harry has a strong sense of justice, and when he believes he's being wrongly punished, he'll be resentful.
In some fanfictions, Snape might punish Ginny for misbehaving in Potions class. For example, she could be given detention for accidentally causing an explosion during a brewing experiment.
Yes, there might be such fanfictions. In the vast world of Hogwarts fanfiction, various scenarios are explored. However, the nature of the 'punishment' can vary greatly from one fanfic to another. It could be related to Draco's misbehavior in the context of Hogwarts' rules, like getting detention or losing house points. Some fanfictions might focus on character development through these interactions between Snape and Draco.
One possible reason could be that Snape fails to deliver crucial information. Voldemort is a very demanding and unforgiving master. If Snape doesn't give him accurate details about the Order of the Phoenix's plans, for example, Voldemort would surely be enraged and punish him.
Snape might endure the punishment in stoic silence. He is a complex character and often used to hiding his true emotions. So, he could just grit his teeth and bear it, all the while plotting his revenge or a way to get out of the situation.