Hun hænger sin våde regnjakke ved døren og sætter hjelmen på hylden i gangen.

Because sin is the reflexive possessive form. It is used when the thing belongs to the subject of the same clause.

• Hun hænger sin våde regnjakke ... = She hangs her own wet raincoat ...

If you said hendes, it would normally mean the raincoat belongs to some other woman, not to the subject hun.

A useful rule:

• sin/sit/sine = the subject owns it
• hans/hendes/deres = someone else owns it

So here sin shows that the woman is hanging up her own raincoat.

These forms agree with the noun being possessed:

• sin → common gender singular
• sit → neuter singular
• sine → plural

Here the noun is regnjakke, which is a common gender singular noun:

• en regnjakke
• therefore: sin regnjakke

Examples:

• sin jakke
• sit hus
• sine sko

So the choice depends on the noun, not on the owner.

Because adjectives usually take -e in certain positions, including after a possessive like sin.

So:

• en våd regnjakke = an/a wet raincoat
• sin våde regnjakke = her own wet raincoat

This -e form is very common after words like:

• den
• det
• de
• possessives such as min, din, sin, hendes

So våde is the expected form here.

Because after a possessive, Danish normally uses the noun in its indefinite form.

Compare:

• regnjakke = raincoat
• regnjakken = the raincoat

After possessives, Danish usually does not use the definite ending:

• min jakke = my jacket
• hendes cykel = her bicycle
• sin regnjakke = her own raincoat

So sin våde regnjakke is correct, not sin våde regnjakken.

These are different prepositions showing different kinds of location:

• ved døren = by/near the door
• på hylden = on the shelf
• i gangen = in the hallway

So Danish uses different prepositions depending on the spatial relationship, just as English does.

A quick breakdown:

• ved = at/by/next to
• på = on
• i = in

These are very common, but they do not always match English one-for-one, so they are worth learning as part of whole phrases.

Because these are definite singular forms, meaning the door, the helmet, the shelf, and the hallway.

In Danish, definiteness is often shown by adding an ending to the noun:

• en dør → døren
• en hjelm → hjelmen
• en hylde → hylden
• en gang → gangen

This is one of the big differences from English. English uses a separate word the, but Danish often attaches the definiteness to the noun itself.

Because Danish can leave out the subject in the second part when it is the same as the first one.

So:

• Hun hænger ... og sætter ...

means:

• She hangs ... and puts ...

This is completely natural because both actions have the same subject: hun.

You could repeat it:

• Hun hænger sin våde regnjakke ved døren, og hun sætter hjelmen på hylden i gangen.

But that sounds a bit more explicit and less smooth in this kind of sentence.

Yes. Both are in the present tense.

• at hænge → hænger
• at sætte → sætter

Danish present tense is usually formed with -r, so these mean:

• hænger = hangs / is hanging
• sætter = puts / is putting / places

The exact English translation depends on context, but grammatically they are simple present forms in Danish.

Because the verbs reflect the kind of action being done.

• at hænge noget = to hang something
• at sætte noget = to put/place/set something, often in an upright or placed position

A raincoat is naturally something you hang.
A helmet is naturally something you put/place on a shelf.

So the verbs are chosen by the object and the situation, much like in English.

Yes. Regnjakke is a compound noun:

• regn = rain
• jakke = jacket

Together:

• regnjakke = raincoat / rain jacket

Danish uses compound nouns very often, usually written as one word. English learners sometimes want to separate them, but in Danish they are normally combined.

Other examples:

• cykelhjelm = bicycle helmet
• soveværelse = bedroom
• toothbrush-type compounds are also common in Danish

So regnjakke is a very typical Danish compound.

Gangen means the hallway or the corridor, but in everyday Danish, gang is a very common word for a hall or passage inside a home.

• i gangen = in the hallway

Korridor also exists, but it often sounds more like a corridor in a school, office, or institution. In a home, gangen is usually the more natural choice.

So this sentence sounds very natural for a house or apartment.

Danish follows the V2 rule in main clauses: the finite verb usually comes in the second position.

So if you start with something else, the verb still comes second, and the subject moves after it.

Original:

• Hun hænger sin våde regnjakke ved døren ...

With a time expression:

• Så hænger hun sin våde regnjakke ved døren ...
• Derefter sætter hun hjelmen på hylden i gangen.

Notice:

• Så hænger hun ... not
• Så hun hænger ...

This is a very important feature of Danish word order.

Yes, in hjelm and hjelmen, the h is silent.

So hjelm begins more like yelm than hyelm.

That is common in Danish in words beginning with hj-:

• hjelm
• hjul
• hjælpe

For English speakers, the spelling can look misleading, so it helps to memorize that hj- usually has a silent h.