Learn Hardcore Croatian: Nemam sitan kusur za kavu. - I don’t have small change for coffee.

In Croatian, the negative particle ne merges with some very common verbs into a single word. With imati (to have), ne + imam becomes nemam.
Other persons: nemaš, nema, nemamo, nemate, nemaju.

What exactly is the grammar of nemam (person/number)?

Nemam is 1st person singular present tense of (ne) imati: (I) don’t have.
If you were addressing someone: Nemaš sitan kusur... = You don’t have...

Why is it sitan kusur—shouldn’t the object be in the accusative case?

It is accusative in function (direct object), but for masculine inanimate singular nouns the accusative form is the same as the nominative.
So kusur looks the same in nominative and accusative, and so does the adjective sitan:

nominative: sitan kusur
accusative (inanimate): sitan kusur (same form)

I’ve heard Nemam sitnog kusura. Is that more correct than Nemam sitan kusur?

Nemam sitnog kusura is very common and often sounds more natural because Croatian frequently uses the genitive with negation and/or a “partitive” meaning (I don’t have any (of it)).
So you’ll encounter both patterns:

Nemam sitan kusur. (direct-object style; understandable)
Nemam sitnog kusura. (very idiomatic: I don’t have any small change)

What does kusur mean exactly, and how is it used in everyday speech?

Kusur means change, especially coins/small change used to pay or to break a bill. It’s often treated like a mass noun (not focusing on exact pieces).
Common phrases:

Imate li sitnog? / Imate li kusura? = Do you have (small) change?
Nemam kusura. = I don’t have change.

What’s the difference between kusur, sitniš, and sitno?

All relate to small money, but with slightly different flavor:

kusur = change (often in the “I need change to pay” sense)
sitniš = small coins / loose change (more “coins in your pocket”)
sitno (noun-like) = small change (colloquial shorthand), e.g. Nemam sitno.

Why is it za kavu and not za kava?

Because za usually requires the accusative when it means for (purpose/aim).
kava (coffee) is feminine; its accusative singular is kavu:

nominative: kava
accusative: kavu So za kavu = for coffee (e.g., to pay for a coffee).

Does za kavu mean “to buy coffee,” “to pay for coffee,” or “for a coffee (drink)”?

Most naturally in this sentence it means to pay for a coffee—you don’t have small change suitable for the price.
Depending on context, za kavu can also mean for coffee more generally (e.g., money set aside for coffee), but in a café situation it’s typically about paying.

Could I also say Nemam sitan kusur za kafu?

Yes, but it depends on region/standard:

kava is standard Croatian.
kafa is more common in Serbian and also appears in some Croatian speech, but kava is the safer choice in Croatia.

Is the word order fixed? Could I say Za kavu nemam sitan kusur?

Croatian word order is flexible. You can move phrases for emphasis:

Nemam sitan kusur za kavu. (neutral)
Za kavu nemam sitan kusur. (emphasizes for coffee specifically)
Sitan kusur nemam za kavu. (emphasizes small change)

How would I make it sound more polite or more “real-life” in a café?

You can soften it with a polite lead-in:

Oprostite, nemam sitnog (kusura) za kavu. = Excuse me, I don’t have small change for the coffee.
Imate li možda sitnog? = Do you happen to have some small change?
Mogu li platiti karticom? = Can I pay by card?

How is Nemam sitan kusur za kavu pronounced (roughly)?

A practical approximation for an English speaker:

Nemam ≈ NEH-mahm
sitan ≈ SEE-tahn
kusur ≈ KOO-soor
za ≈ zah
kavu ≈ KAH-voo
Croatian stress is fairly even compared to English; aim for clear vowels and rolled/light r in kusur if you can.