Min datter hænger forklædet på stolen, før hun går ud i haven igen.

After a possessive like min, Danish normally uses the noun in the indefinite form:

• min datter = my daughter
• min bil = my car
• vores hus = our house

So datteren means the daughter, but after min you say min datter, not min datteren.

Those endings show the definite form of the noun, and the ending depends on the noun’s gender.

• et forklæde = an apron
  forklædet = the apron
• en stol = a chair
  stolen = the chair

So:

• -et is the definite ending for many neuter nouns
• -en is the definite ending for many common gender nouns

This is one of the first big differences from English: Danish often puts the on the end of the noun instead of using a separate word.

Yes, hænger is from hænge, and here it means something like hangs or hangs up.

The choice of verb reflects how the object is placed. Danish often cares about position:

• lægge = lay something down
• stille/sætte = set something somewhere
• hænge = hang something

So hænger forklædet på stolen suggests she puts the apron so that it is hanging on the chair, not just lying there.

Because -r is the normal ending for the present tense in Danish.

• hænge → hænger
• gå → går

Unlike English, Danish verbs do not change form depending on the subject:

• jeg går
• du går
• hun går
• vi går

They all use the same present-tense form.

Because før hun går ud i haven igen is a subordinate clause.

In Danish, there are two accepted comma styles:

• one that does put a comma before subordinate clauses
• one that does not

So both of these may be seen:

• Min datter hænger forklædet på stolen, før hun går ud i haven igen.
• Min datter hænger forklædet på stolen før hun går ud i haven igen.

The version with the comma is very common in teaching materials because it makes the clause structure clearer.

Because før introduces a subordinate clause, and subordinate clauses do not use the normal Danish verb-second pattern.

In the main clause, Danish usually puts the finite verb in second position:

• Min datter hænger forklædet på stolen

But after før, the order is more like:

• før + subject + verb
• før hun går ud i haven igen

If you move the før-clause to the front, the main clause then shows normal inversion:

• Før hun går ud i haven igen, hænger min datter forklædet på stolen.

Går ud i haven means goes out into the garden.

The parts do different jobs:

• går = goes/walks
• ud = out
• i haven = into/in the garden

So ud is important because it shows movement from inside to outside.

Compare:

• går ud i haven = goes out into the garden
• går i haven = walks in the garden / is in the garden walking around

And til haven would usually not be the natural choice here. Til is more like to a destination in a general sense, but with areas like haven, Danish commonly uses ud i.

In this sentence, it can cover the idea of onto the chair as part of the action, even though the preposition is still på.

Danish does not always separate on and onto the way English does. The verb and context often tell you whether the meaning is:

• location: on
• movement toward that location: onto

So hænger forklædet på stolen naturally means she places it so that it ends up on the chair.

That is a very natural place for igen in this kind of clause.

• hun går ud i haven igen

Here igen means again, and it applies to the whole action of going out into the garden another time.

In Danish, adverbs like this are often placed late in the clause, especially when they modify the whole event rather than just one single word.

Because Danish often uses the present tense for future meaning when the time relationship is already clear from the context.

Here før already shows that the going-out happens after the hanging action, so Danish does not need a special future form.

English does something similar:

• before she goes out
• not usually before she will go out

So før hun går ud i haven igen is completely normal Danish.