Jeg hoster om natten, når luften er kold.

Hoster is the present tense form of the verb at hoste = to cough.

• at hoste = infinitive, to cough
• hoster = present tense, cough / am coughing

So Jeg hoster means I cough or I am coughing, depending on context.

A useful pattern:

• jeg hoster = I cough
• jeg hostede = I coughed
• jeg har hostet = I have coughed

It can mean either, depending on context.

In this sentence, it most naturally sounds like a general or repeated situation:

Jeg hoster om natten, når luften er kold.
= I cough at night when the air is cold.

That suggests something that happens regularly or whenever that condition is true, not just one single moment right now.

Danish present tense often covers both:

• I cough
• I am coughing

Context tells you which one is meant.

Om natten is the normal Danish expression for at night in a general, habitual sense.

So:

• om natten = at night, during the night, nights in general

This is different from English, where at night uses at. Danish uses om in many time expressions of this kind.

Other examples:

• om dagen = during the day / in the daytime
• om morgenen = in the morning
• om vinteren = in the winter

So you should learn om natten as a fixed, very common expression.

No. In this sentence, om does not mean about.

Danish om has several meanings depending on context. Here it is used in a time expression, and it means something like:

• during
• in
• at (depending on how English phrases it)

So om natten is best understood as at night or during the night.

This is a good reminder that Danish prepositions do not always match English ones directly.

Når is the right choice here because the sentence describes something that happens repeatedly or whenever a condition is true.

• når = when / whenever
• da = when, but usually about one specific time in the past
• hvis = if

In this sentence, the meaning is: Whenever the air is cold, I cough at night.

So når fits best.

Compare:

• Jeg hoster, når luften er kold. = I cough when/whenever the air is cold.
• Jeg hostede, da luften blev kold. = I coughed when the air got cold.
  (one past event)
• Jeg hoster, hvis luften er kold. = I cough if the air is cold.
  (more conditional, slightly less natural here)

Luften is the definite form of luft.

• luft = air
• luften = the air

In Danish, the definite article is often attached to the end of the noun:

• en luft is not normally used in this meaning
• luften = the air

Here, Danish prefers luften because it refers to the air in the surrounding environment, roughly the air as we experience it.

A useful contrast:

• Der er ikke nok luft herinde. = There is not enough air in here.
  (luft = air as a substance)
• Luften er kold. = The air is cold.
  (luften = the air around us)

Because når luften er kold is a subordinate clause, and Danish word order is different in subordinate clauses.

In a normal subordinate clause, the order is: subject + verb

So:

• når luften er kold = when the air is cold

Not:

• når er luften kold

The version with the verb before the subject would look more like a main-clause question pattern, which is not what we want here.

A helpful extra point: if you add ikke, you can clearly see the subordinate-clause pattern:

• når luften ikke er kold = when the air is not cold

In subordinate clauses, words like ikke usually come before the finite verb.

Because the adjective agrees with the subject, and luften is common gender singular.

• en luft would be common gender
• so the adjective is kold

Compare:

• Luften er kold. = The air is cold.
  (common gender singular)
• Vejret er koldt. = The weather is cold.
  (vejret is neuter, so koldt)
• Dagene er kolde. = The days are cold.
  (plural, so kolde)

Since luften is singular common gender, kold is correct.

Yes. Danish allows that, and it is very natural:

Når luften er kold, hoster jeg om natten.

This means the same thing, but the focus shifts slightly. Starting with the når-clause gives more emphasis to the condition.

Notice what happens in the main clause after a fronted clause:

• Når luften er kold, hoster jeg om natten.

Here, hoster comes before jeg. That is normal Danish V2 word order in main clauses: when something else comes first, the finite verb comes next.

No, they are different.

• om natten = at night, during the night, nights in general
• i nat = tonight / last night, depending on context

So in your sentence:

• Jeg hoster om natten means I cough at night as a general pattern.

But:

• Jeg hoster i nat would mean something connected to this night, not a general habit.

That is an important distinction:

• general repeated time -> om natten
• one specific night -> i nat