Im Kurs bestimmen wir zuerst den Kasus.

Im is a contraction of in dem.

• in = in
• dem = the (dative masculine/neuter)

So:

• im Kurs = in dem Kurs

German very often contracts certain preposition + article combinations:

• in dem → im
• an dem → am
• zu dem → zum
• bei dem → beim

In this sentence, im Kurs means in the course / in class.

Because here in expresses location, not movement.

German uses different cases after some prepositions, including in:

• dative for location: where?
• accusative for direction/movement: where to?

So:

• im Kurs = in the course / in class → location → dative
• in den Kurs = into the course → movement/direction → accusative

In your sentence, the idea is that this happens in the course/class, so German uses the dative: im Kurs.

Because German main clauses follow the V2 rule: the finite verb must be in second position.

Here the first position is occupied by Im Kurs, so the verb comes next:

• Im Kurs | bestimmen | wir | zuerst | den Kasus

Even though wir is the subject, it does not have to come first. If something else is placed first, the subject usually comes after the verb.

Compare:

• Wir bestimmen zuerst den Kasus im Kurs.
• Im Kurs bestimmen wir zuerst den Kasus.

Both are grammatical, but the second one emphasizes Im Kurs.

This is a result of the same V2 word order rule.

In English, the subject usually stays before the verb. In German, if another element is moved to the front of a main clause, the finite verb still has to stay in second position, and the subject often moves after it.

So:

• normal order: Wir bestimmen zuerst den Kasus.
• with fronted phrase: Im Kurs bestimmen wir zuerst den Kasus.

This is often called inversion in English-language explanations of German word order.

Yes, bestimmen basically means to determine, to identify, or to decide depending on context.

In a grammar/classroom context, den Kasus bestimmen usually means:

• to identify the case
• to determine which case it is

So here it does not mean something like to command. It means analyzing grammar and figuring out whether something is nominative, accusative, dative, or genitive.

Because Kasus is the direct object of bestimmen, and bestimmen takes the accusative.

The noun Kasus is masculine:

• nominative: der Kasus
• accusative: den Kasus
• dative: dem Kasus
• genitive: des Kasus

Since the sentence says we determine the case, the case is the thing being determined, so it appears in the accusative:

• den Kasus

Kasus is the grammatical term for case.

In German grammar, Kasus refers to forms like:

• Nominativ
• Akkusativ
• Dativ
• Genitiv

So den Kasus bestimmen means identifying which grammatical case a noun, pronoun, or article has in a sentence.

Zuerst means first or first of all.

It tells you the order of actions:

• Im Kurs bestimmen wir zuerst den Kasus. = In class, we first identify the case.

It is an adverb of sequence/time. It often appears in positions like this:

• Wir bestimmen zuerst den Kasus.
• Zuerst bestimmen wir den Kasus.

Both are fine; the placement changes the emphasis slightly.

Yes. It is grammatically correct.

• Im Kurs is the normal contracted form.
• In dem Kurs is the full form.

In everyday German, im Kurs sounds more natural in most cases.

Using the full form in dem Kurs may sound a bit more explicit or contrastive, for example if you are distinguishing one course from another.

Because they are nouns.

In German, all nouns are capitalized, not just proper names. So:

• der Kurs
• der Kasus

This is one of the most noticeable spelling differences between German and English.

It can mean both, depending on context.

Kurs often refers to:

• a course
• a class
• a structured set of lessons

So im Kurs can often be translated naturally as:

• in the course
• in class

The best English choice depends on the context, but the German word itself is completely normal here.

Yes, very strongly.

The phrase den Kasus bestimmen is standard grammar/classroom language. It suggests that people in the course are analyzing a sentence and identifying the grammatical case of a word or phrase.

So even without extra context, Kasus makes the sentence sound clearly grammatical or linguistic rather than general everyday speech.