Min søster betaler sin gæld langsomt af, så hun stadig har penge til huslejen.

Because sin/sit/sine is the reflexive possessive in Danish. It refers back to the subject of the same clause.

Here, the subject is Min søster, so sin gæld means her own debt.

If you said hendes gæld, it would normally mean another woman’s debt, not the sister’s own debt.

Also, the form sin is used because gæld is a common-gender singular noun:

• sin gæld
• sit hus
• sine penge

Because possessives like min, din, sin, hendes already act as determiners.

So Danish says:

• min søster
• sin gæld

not:

• en min søster
• en sin gæld

This works much like English: you say my sister, not the my sister.

This is the normal V2 word order of Danish main clauses: the finite verb usually comes in second position.

So:

• Min søster = first element
• betaler = second element

That is why the clause begins: Min søster betaler ...

You could also move another element to the front, but then the verb would still stay second:

• Langsomt betaler min søster sin gæld af.

Because this is a particle verb: betale af or closely related afbetale, meaning pay off / pay down.

In Danish, particles like af, op, ud, med are often separated from the finite verb in a main clause.

So you get:

• Hun betaler gælden af.

rather than keeping the whole thing together.

That is very natural Danish word order.

Yes. Min søster afbetaler sin gæld langsomt is also correct and natural.

Both versions are very close in meaning:

• betaler sin gæld ... af
• afbetaler sin gæld ...

In everyday Danish, both patterns are common. A learner should understand both.

Because langsomt is an adverb, and it describes how she pays.

• langsom = slow, a form used with nouns
• langsomt = slowly, used with verbs

Here it modifies betaler, so Danish uses the adverb form:

• Hun betaler langsomt.

This is very common:

• hurtigt = quickly
• pænt = nicely
• langsomt = slowly

Here så means something like so that or so in the sense of with the result/purpose that.

So the sentence is saying she pays off her debt slowly so that she still has money for the rent.

This is not quite the same as simple therefore.

Because after så here, Danish uses subordinate clause word order.

In a subordinate clause, sentence adverbs such as stadig, ikke, nok, jo usually come before the finite verb.

So:

• main clause: Hun har stadig penge.
• subordinate clause: ... så hun stadig har penge.

This is one of the most important word-order patterns in Danish.

Because huslejen is the definite form: the rent.

The ending -en marks definite singular for a common-gender noun:

• husleje = rent
• huslejen = the rent

In this sentence, it refers to a specific, understood rent payment, so the definite form sounds natural.

Danish often does this with familiar recurring expenses:

• regningen = the bill
• huslejen = the rent

Because penge usually means money in general, not the specific money.

So:

• penge = money
• pengene = the money

In this sentence, she still has money available, so penge is the natural choice.

Also, penge is normally used as a plural-form noun in this meaning, so you do not say en penge here. If you want to make it less general, you could say:

• nogle penge = some money