Po alarmie sprawdziliśmy mieszkanie i okazało się, że w kuchni nie było pożaru, tylko dużo dymu z piekarnika.

Because the preposition po meaning after requires the locative case in this use.

• alarm = base form
• po alarmie = after the alarm

So this is a normal case pattern:

• po obiedzie = after dinner
• po pracy = after work
• po filmie = after the movie

Here, alarmie is the locative singular form of alarm.

Sprawdziliśmy is the 1st person plural past tense form of sprawdzić = to check.

It means we checked.

Breakdown:

• sprawdzi- = verb stem
• -liśmy = past tense ending for we

This verb is perfective, so sprawdziliśmy suggests a completed action: we checked the apartment and finished checking it.

Compare:

• sprawdzaliśmy = we were checking / we used to check
• sprawdziliśmy = we checked

In this sentence, the completed action makes sense.

Because mieszkanie is the direct object of sprawdziliśmy, so it is in the accusative case. But for many neuter nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: mieszkanie
• accusative: mieszkanie

That is why the form does not visibly change.

Okazało się is a very common Polish expression meaning it turned out.

The verb is okazać się = to turn out / prove to be.

Why okazało?

• It is in the past tense
• It is in the neuter singular form because Polish often uses a neuter form in impersonal statements like it turned out that...

Why się?

• It is part of the verb itself: okazać się
• You usually learn it as one unit

So:

• okazało się, że... = it turned out that...

This is one of the most useful fixed patterns in Polish.

Że means that and introduces a subordinate clause.

So:

• okazało się, że... = it turned out that...

Everything after że gives the content of what turned out to be true.

This is very similar to English that, although Polish uses że very regularly in this structure.

Because w meaning in usually takes the locative case when it refers to location.

• kuchnia = kitchen
• w kuchni = in the kitchen

So this is another normal case pattern:

• w domu = in the house
• w szkole = at school / in the school
• w kuchni = in the kitchen

This is a very important Polish pattern.

When Polish says there is / there was not, it normally uses:

• nie ma
  • genitive
• nie było
  • genitive

So:

• był pożar = there was a fire / a fire existed
• nie było pożaru = there was no fire

Here:

• pożar is nominative
• pożaru is genitive singular

This genitive after negated existence is very common:

• Nie ma czasu. = There is no time.
• Nie było problemu. = There was no problem.
• Nie ma wody. = There is no water.

So nie było pożaru is the correct structure.

Here tylko means only / just / rather in a contrast:

• nie było pożaru, tylko dużo dymu
  = there was no fire, only a lot of smoke

It corrects the previous idea:

• not a fire
• just a lot of smoke

Now for dużo dymu:

dużo means a lot of and it requires the genitive case.

So:

• dym = smoke
• dymu = genitive singular

Similar examples:

• dużo czasu = a lot of time
• dużo pracy = a lot of work
• dużo ludzi = a lot of people

Because z here means from / out of, and in this meaning it takes the genitive case.

• piekarnik = oven
• z piekarnika = from the oven / out of the oven

So the smoke was coming from the oven.

Compare:

• z domu = from the house
• z kuchni = from the kitchen
• z piekarnika = from the oven

Because the sentence tells a short sequence of completed events in the past:

• sprawdziliśmy = we checked
• okazało się = it turned out
• nie było = there was not

This is very natural in Polish narrative style. The sentence first gives the action, then the result, then the specific finding.

Yes, Polish often does not use articles because Polish has no words for a/an/the.

So po alarmie can mean:

• after the alarm
• after an alarm

Which one is meant depends on context. In this sentence, English would probably use the because the alarm is a specific event already understood in the situation.

Yes, Polish word order is fairly flexible, but this version sounds natural and clear.

Current structure:

• Po alarmie = sets the time first
• sprawdziliśmy mieszkanie = tells the action
• i okazało się, że... = gives the result
• w kuchni nie było pożaru, tylko dużo dymu z piekarnika = gives the important detail

You could move parts around, but the original order is very normal because it presents the information in a logical sequence:

1. after the alarm
2. we checked
3. it turned out that
4. the problem was smoke, not fire

This is a useful vocabulary question.

• pożar = a fire in the sense of a blaze / a fire incident, especially dangerous or accidental
• ogień = fire as a substance or flame

So in this sentence:

• nie było pożaru = there was no fire incident

If you said ogień, it would suggest actual flames rather than the whole event of a fire. Since the sentence contrasts fire with smoke, pożar is the more natural word here.