Learn Russian: Нельзя наказывать ребёнка за каждую ошибку. - One must not punish a child for every mistake.

The preposition за is used here to show the reason or cause for an action, which in this case is punishment. Just like when you say 'thank you FOR something' or 'pay FOR something', you use за plus the Accusative case to show what the action is responding to.

Why are we using the imperfective verb наказывать instead of the perfective наказать?

When нельзя is followed by an imperfective verb, it means 'one must not' or 'it is forbidden' to do something. If you used the perfective verb наказать, it would change the meaning of нельзя to 'it is impossible' to punish them.

What case is ребёнка and why does it look like the Genitive case?

It is in the Accusative case because the child is the direct object receiving the punishment. For animate masculine nouns (people and animals), the Accusative form is identical to the Genitive form. Notice also that the letter 'о' in ребёнок drops out when adding the ending.

What case is каждую ошибку in, and why?

They are in the Accusative case because they follow the preposition за. Since ошибка is a feminine noun, its Accusative ending changes to -у, and the matching adjective каждая changes to каждую.