Po wejściu do wagonu pytam, czy ten przedział jest wolny.

Because po in the sense of after takes the locative case.

• base form: wejście = entering / entrance
• locative singular: wejściu

So po wejściu literally means after the entering, but in natural English it is simply after entering.

This is a very common Polish pattern:

• po przyjeździe = after arriving
• po powrocie = after returning
• po skończeniu pracy = after finishing work

In this sentence, wejście is a verbal noun, formed from wejść = to enter.

Because do normally requires the genitive case.

• base form: wagon
• genitive singular: wagonu

So:

• do wagonu = into the carriage / train car

This is the standard pattern:

• do domu = into/to the house
• do pokoju = into/to the room
• do pociągu = into/to the train

Not really. In Polish, wagon commonly means a railway car / carriage, not the English idea of a farm wagon or covered wagon.

So in this sentence:

• wagon = a train car / carriage

Depending on the variety of English, the best translation is often:

• carriage in British English
• car or railcar in American English

These two expressions are close, but they are not identical.

• po wejściu do wagonu = after entering the carriage
• wchodząc do wagonu = while entering / as I enter the carriage

So po wejściu shows that one action is completed first, and then the next action happens.

Sequence:

1. I enter the carriage.
2. Then I ask.

With wchodząc do wagonu, the asking would sound as if it happens during the act of entering.

Because Polish usually drops subject pronouns when they are not needed.

The verb ending already tells you who the subject is:

• pytam = I ask

So ja is unnecessary unless you want emphasis or contrast.

For example:

• Pytam = I ask
• Ja pytam = I ask / I’m the one asking

In neutral Polish, leaving out ja is more natural.

This is mainly about aspect and tense.

• pytać is imperfective
• zapytać is perfective

So:

• pytam = I ask / I am asking
• zapytam = I will ask

A perfective verb normally does not have a true present tense in Polish. Its present-looking forms usually refer to the future.

That means:

• pytam works for a present-time narration, habitual situation, or vivid storytelling
• zapytam would usually mean I will ask

If you wanted a simple completed past action, you would more likely say:

• Po wejściu do wagonu zapytałem, czy ten przedział jest wolny.

Polish often uses the present tense narratively, just like English can in storytelling:

• I walk in, I look around, and I ask...

So Po wejściu do wagonu pytam... can sound like:

• a vivid narration of events
• a habitual situation
• a description of what I typically do in that situation

If the speaker wanted a plain past statement, they would usually switch to the past tense:

• Po wejściu do wagonu zapytałem...

Here czy introduces an indirect yes/no question.

So:

• pytam, czy ten przedział jest wolny = I ask whether / if this compartment is free

Compare:

• direct question: Czy ten przedział jest wolny?
• indirect question: Pytam, czy ten przedział jest wolny.

So after verbs like ask, know, check, see, Polish often uses czy where English uses if or whether.

Because czy ten przedział jest wolny is a subordinate clause.

Polish punctuation normally separates the main clause from a subordinate clause with a comma:

• Pytam, czy...
• Nie wiem, czy...
• Sprawdzam, czy...

So the comma here is standard and expected.

Przedział here means a train compartment: an enclosed section of a carriage with seats inside.

So this is not just any seat or area. It suggests a train with separate compartments.

That matters because in a modern open-plan carriage, a Polish speaker might ask something different, for example:

• Czy to miejsce jest wolne? = Is this seat free?
• Czy tu wolne? = Is this free here?

But with przedział, the speaker is asking about the whole compartment.

Because both ten and wolny have to agree with przedział.

Przedział is:

• masculine
• singular
• nominative

So the matching forms are:

• ten przedział
• wolny

Compare:

• ta kabina jest wolna = this cabin is free
• to miejsce jest wolne = this place/seat is free

This is basic Polish agreement: demonstratives and adjectives match the noun in gender, number, and case.

Here wolny means free, vacant, or unoccupied.

So czy ten przedział jest wolny means:

• is this compartment free?
• is this compartment vacant?
• is nobody sitting here?

This is a very common Polish use of wolny for places, seats, tables, compartments, and similar things.

Examples:

• To miejsce jest wolne. = This seat is free.
• Ten stolik jest wolny. = This table is free.

Yes, the given word order is natural.

• Po wejściu do wagonu sets the scene first
• pytam gives the main action
• czy ten przedział jest wolny gives the content of the question

Polish word order is flexible, so other versions are possible, but they may change the emphasis.

For example:

• Po wejściu do wagonu pytam, czy ten przedział jest wolny.
  Neutral and natural.

• Pytam po wejściu do wagonu, czy ten przedział jest wolny.
  Possible, but it puts more focus on when I ask.

So the original version is a good, natural narrative order.