Moram da pitam kasirku koliko košta ovaj kaput.

Because Serbian often drops subject pronouns when the verb ending already makes the subject clear.

• moram = I must / I have to
• The ending -am already tells you it is 1st person singular

So Ja moram da pitam... is possible, but it usually adds emphasis, like I have to ask.

Here da introduces a verb phrase that works a lot like an infinitive in English.

• moram = I must
• da pitam = literally something like that I ask, but in natural grammar it means to ask

So moram da pitam means I must ask.

This da + present tense pattern is extremely common in Serbian, especially after modal verbs like:

• moram da... = I must...
• mogu da... = I can...
• želim da... = I want to...

Yes, it is understandable and grammatically possible, but in everyday Serbian moram da pitam sounds more natural.

So:

• Moram da pitam = the most usual everyday Serbian wording
• Moram pitati = possible, but less typical in modern Serbian speech

A learner will sound more natural using da + present here.

Because kasirku is the accusative singular form, and it is the direct object of pitam.

Base form:

• kasirka = cashier (female), nominative singular

After pitati:

• pitam kasirku = I ask the cashier

So the ending changes because Serbian marks grammatical case.

Yes. Kasirka is feminine.

If the cashier is male, you use kasir.

Then the sentence would be:

• Moram da pitam kasira koliko košta ovaj kaput.

Notice the accusative form:

• kasir → kasira

because masculine animate nouns usually take the accusative form that matches the genitive.

Because ovaj kaput is the subject of košta, so it stays in the nominative.

Inside the clause:

• koliko košta ovaj kaput = how much this coat costs

Here:

• košta = costs
• ovaj kaput = this coat, the thing doing the costing

So it is not the object there; it is the subject of košta.

Also, ovaj has to agree with kaput:

• masculine
• singular
• nominative

That is why you get ovaj kaput.

Yes, both are possible.

Serbian word order is more flexible than English word order.
The version in your sentence, koliko košta ovaj kaput, is very natural and common.

Both of these work:

• koliko košta ovaj kaput
• koliko ovaj kaput košta

Unlike English, Serbian does not need does-support or strict inversion.
So English how much does this coat cost? becomes simply:

• Koliko košta ovaj kaput?

The embedded question keeps the same basic pattern.

Here it means how much.

With prices, koliko is the normal word:

• Koliko košta? = How much does it cost?
• Koliko je? = How much is it?

The same word koliko can also mean how many, depending on context, but with košta it clearly means how much.

Because its subject is ovaj kaput, which is singular.

• ovaj kaput = this coat
• one coat → singular
• so the verb is singular: košta

If it were plural, the verb would change:

• Ovi kaputi koštaju... = These coats cost...

So košta agrees with kaput.

Good question. Serbian aspect does not work exactly like English one time vs many times.

Pitati is imperfective, but it is very commonly used as the neutral everyday verb to ask.
So moram da pitam sounds completely normal.

A perfective verb like upitati can also exist, but it often sounds more marked or less neutral in everyday speech.

So for a learner, the safest natural choice is:

• Moram da pitam...

Because Serbian does not have articles.

English says:

• the cashier
• this coat

Serbian usually expresses that meaning through context, word order, or demonstratives like ovaj.

So in this sentence:

• kasirku does not need a word for the
• ovaj kaput already means this coat, so it is naturally definite

That is normal Serbian grammar, not something missing.