Welcher Kasus passt hier? Die Dozentin erklärt, wann der Dativ und wann der Akkusativ benutzt wird.

Because Kasus is the subject of the question.

The verb is passt (fits), and we ask: Which case fits here?
So welcher is in the nominative masculine singular to match der Kasus.

• der Kasus → nominative
• welcher Kasus → which case

You would use welchen Kasus if Kasus were in the accusative, for example after a verb that takes a direct object:

• Welchen Kasus benutzt man hier? = Which case do you use here?

Because Die Dozentin is the subject of the sentence, so it must be in the nominative.

• die Dozentin = nominative singular
• der Dozentin = dative or genitive singular

Here, the lecturer explains, so the lecturer is doing the action. That makes it the subject:

• Die Dozentin erklärt ...

This is an embedded clause after erklärt. The sentence means that the lecturer explains when the dative is used and when the accusative is used.

German often uses wann in indirect questions like this:

• Sie erklärt, wann ... benutzt wird.
• Ich weiß nicht, wann ...
• Kannst du sagen, wann ...?

So wann introduces the idea of when one case is used and when the other is used.

Because the part after erklärt is a subordinate clause / indirect question, and in German, the conjugated verb usually goes to the end of such clauses.

Main clause:

• Die Dozentin erklärt ...

Subordinate clause:

• ..., wann der Dativ und wann der Akkusativ benutzt wird.

That final wird belongs to the passive form benutzt wird.

Because this is the passive voice.

• Man benutzt den Dativ. = People use the dative.
• Der Dativ wird benutzt. = The dative is used.

So:

• wann der Dativ benutzt wird literally corresponds to
• when the dative is used

German forms the passive with:

• werden
  • past participle

Here:

• wird = form of werden
• benutzt = past participle

Because in the passive construction, they are the subjects of their respective clauses.

The sentence is effectively shorthand for:

• ..., wann der Dativ benutzt wird und wann der Akkusativ benutzt wird.

In each part:

• der Dativ is the subject of wird benutzt
• der Akkusativ is the subject of wird benutzt

So both stay in the nominative.

Because German often leaves out repeated words when the meaning is clear.

Full version:

• Die Dozentin erklärt, wann der Dativ benutzt wird und wann der Akkusativ benutzt wird.

Shorter, more natural version:

• Die Dozentin erklärt, wann der Dativ und wann der Akkusativ benutzt wird.

This kind of omission is common when the repeated part is identical and easy to understand from context.

It is best understood as an abbreviated structure covering two parallel indirect questions:

• wann der Dativ benutzt wird
• wann der Akkusativ benutzt wird

German combines them to avoid repetition:

• wann der Dativ und wann der Akkusativ benutzt wird

So grammatically, it is a coordinated structure with one repeated verb phrase left out.

Yes, but the meaning changes slightly.

• wann der Dativ und wann der Akkusativ benutzt wird = when the dative and when the accusative are used
• wie der Dativ und der Akkusativ benutzt werden = how the dative and accusative are used

So:

• wann focuses on in which situations / under what conditions
• wie focuses on the manner or rules of usage

Also note that with der Dativ und der Akkusativ as a plural subject together, you would say:

• benutzt werden not
• benutzt wird

Because the verb agrees with the understood subject in each part separately, not with der Dativ und der Akkusativ as one plural subject.

The sentence really means:

• wann der Dativ benutzt wird
• und wann der Akkusativ benutzt wird

Each clause has a singular subject:

• der Dativ → wird
• der Akkusativ → wird

If both together were the subject of one clause, then you would use werden:

• Der Dativ und der Akkusativ werden benutzt.

But that is not the structure here.

Here hier means something like here, in this spot, or in this context.

It usually refers to a particular place in an exercise, sentence, or grammar example:

• Welcher Kasus passt hier? = Which case fits here?

So it is not about physical location only; it often means at this point in the sentence.

Yes. passen is very common for meaning to fit or to be appropriate.

In grammar explanations, German speakers often say:

• Welcher Kasus passt hier?
• Welche Präposition passt?
• Welche Form passt hier?

It sounds natural and idiomatic. It means the form is the correct one for that context.

Because the sentence is naming grammatical categories, not using them inside another construction.

When you talk about cases as concepts, German often uses the noun directly with the article:

• der Dativ
• der Akkusativ
• der Nominativ
• der Genitiv

So:

• wann der Dativ benutzt wird means
• when the dative is used

If you said im Dativ, that would mean in the dative, which is possible in other contexts, but it is not the structure used here.

Yes. Fall and Kasus both mean grammatical case.

• der Kasus is more technical or grammatical
• der Fall is also very common and often easier for learners at first

So both are possible:

• Welcher Kasus passt hier?
• Welcher Fall passt hier?

In grammar books, you may see either one.