Vatrogaskinja je otvorila prozor da dim brže izađe van.

Vatrogaskinja means female firefighter.

The ending -inja is a common way to form some feminine nouns in Croatian. Here:

• vatrogasac = firefighter (usually the basic masculine form)
• vatrogaskinja = female firefighter

This matches the verb form otvorila, which is also feminine singular, so the sentence clearly tells us the firefighter is a woman.

Je otvorila is the standard perfect tense in Croatian:

• je = auxiliary verb (is/has-type helper)
• otvorila = past participle, feminine singular

Together, je otvorila means opened / has opened depending on context.

Croatian often uses the perfect tense where English might simply use the past:

• Vatrogaskinja je otvorila prozor = The firefighter opened the window

In everyday speech, the auxiliary can sometimes be omitted in certain styles or contexts, but je otvorila is the normal full form here.

In Croatian, the past participle agrees with the subject in gender and number.

Since vatrogaskinja is feminine singular, the verb must also be feminine singular:

• masculine: otvorio
• feminine: otvorila
• neuter: otvorilo

So:

• Vatrogasac je otvorio prozor. = A male firefighter opened the window.
• Vatrogaskinja je otvorila prozor. = A female firefighter opened the window.

Yes, prozor is in the accusative, because it is the direct object of otvorila.

But for many masculine inanimate nouns in Croatian, the accusative singular looks the same as the nominative singular.

So:

• nominative: prozor = window
• accusative: prozor = window

That is why the form does not visibly change.

Here da means so that or in order that.

The clause da dim brže izađe van expresses purpose:

• Vatrogaskinja je otvorila prozor = The firefighter opened the window
• da dim brže izađe van = so that the smoke would get out faster

So the whole sentence means something like:

• The firefighter opened the window so that the smoke would get out faster.

This is a very common Croatian structure.

Great question. After da in purpose clauses, Croatian commonly uses a form that looks like the present tense, but the whole clause is understood as purpose/intended result, not simple present time.

So:

• izađe literally looks like a present form of a perfective verb
• in this sentence, it means something like get out / would get out

This is normal Croatian grammar:

• Otvorila je prozor da dim izađe. = She opened the window so that the smoke could/would go out.

Also important: izaći is a perfective verb, and its “present” forms often refer to a single completed event in the future or intended outcome, not an ongoing present action.

Because its subject is dim (smoke), which is singular.

In the subordinate clause:

• dim = subject
• izađe = verb, 3rd person singular

So the structure is:

• da dim izađe = so that the smoke gets/goes out

Even though English sometimes treats substances a little differently, dim is just a normal singular noun here.

Brže means faster / more quickly.

It is the comparative form of brzo (fast, quickly).

• brzo = quickly
• brže = more quickly / faster

So:

• da dim brže izađe van = so that the smoke gets out faster

It modifies the verb phrase, not the noun. In other words, it describes how the smoke goes out.

Yes, izaći already means to go out / come out, so van is not strictly necessary.

• izaći = to go out
• van = out, outside

So izađe van is a bit like saying go out outside if translated too literally. But in real Croatian, this is very natural and common. Van adds emphasis and makes the direction especially clear.

You could also hear:

• da dim izađe
• da dim izađe van

Both are normal. The second one sounds a bit more explicit.

Yes. Both main verbs are from perfective verbs:

• otvoriti = to open (successfully, as a completed action)
• izaći = to go out / come out (as a completed event)

That fits the meaning well:

• the firefighter opened the window as a completed action
• the smoke is intended to get out as a completed result

If you used the imperfective partners, the meaning would shift more toward process or repeated action:

• otvarati = to be opening / open repeatedly
• izlaziti = to be going out / go out repeatedly

Here, the sentence is about one concrete completed act with a clear goal, so perfective is the natural choice.

Yes. Croatian word order is fairly flexible, though not completely free. This sentence is neutral and natural:

• Vatrogaskinja je otvorila prozor da dim brže izađe van.

But you could also hear variations depending on emphasis, for example:

• Prozor je otvorila vatrogaskinja da dim brže izađe van.
  Emphasis on the window
• Da dim brže izađe van, vatrogaskinja je otvorila prozor.
  Emphasis on the purpose

The original version is probably the most straightforward for a learner.

Yes, often you can.

For example:

• Vatrogaskinja je otvorila prozor da dim brže izađe van.
• Vatrogaskinja je otvorila prozor kako bi dim brže izašao van.

Both mean roughly:

• The firefighter opened the window so that the smoke would get out faster.

The difference is mostly stylistic:

• da is very common and straightforward
• kako bi can sound a bit more formal or explicit

Notice that with kako bi, the verb form changes:

• izađe with da
• izašao with kako bi

Both patterns are useful to learn.