Hun vil gerne have et mere moderne køkken i sin lejlighed.

Because gerne softens vil and makes it sound more like would like to rather than a bare wants to.

• hun vil have ... = she wants to have ...
• hun vil gerne have ... = she would like to have ...

So gerne often adds a polite, natural, less blunt tone. In everyday Danish, vil gerne is very common.

This is the normal word order in Danish.

The finite verb comes early in the clause, so you get:

• Hun vil gerne have ...

Here:

• vil = the finite verb
• gerne = an adverb
• have = the infinitive/main verb

With modal verbs like vil, kan, skal, må, the main verb usually stays in the infinitive at the end of the verb group:

• Hun vil have ...
• Hun kan købe ...
• Hun skal male ...

And adverbs like gerne often sit between the modal and the infinitive:

• vil gerne have

Because have is the infinitive form, and after a modal verb like vil, Danish normally uses the infinitive.

So:

• vil = finite/modal verb
• have = infinitive

Compare:

• Hun har et køkken. = here har is the main finite verb
• Hun vil have et køkken. = here have follows the modal vil

This is similar to English:

• She has
• She wants to have

Except Danish usually does not add a separate to after modal verbs.

Because køkken is a neuter noun in Danish, and neuter nouns take et.

Danish nouns have two grammatical genders:

• en words
• et words

So:

• en lejlighed
• et køkken

Unfortunately, noun gender usually has to be learned with the noun. A good habit is to memorize nouns together with their article:

• et køkken
• en stol
• et hus

Because mere moderne is the comparative form: more modern.

• moderne = modern
• mere moderne = more modern

Danish often forms the comparative in two ways:

1. With -ere
   • stor → større
2. With mere
   • especially with longer adjectives or many adjectives of foreign origin
   • moderne → mere moderne

So et mere moderne køkken means a kitchen that is more modern than the current one or than some other standard.

Good question. In this sentence, the adjective is inside a comparative phrase:

• et mere moderne køkken

Here, mere moderne works as a comparative expression, and moderne itself does not take an extra neuter ending.

Also, moderne already ends in -e, so it often looks the same across forms anyway.

Compare:

• et stort køkken = a big kitchen
• et mere moderne køkken = a more modern kitchen

So the important thing here is that the comparison is marked by mere, not by adding a special ending to moderne.

Because sin is the reflexive possessive form. It is used when the possessor is the same as the subject of the clause.

In this sentence:

• subject = Hun
• owner of the apartment = the same woman

So Danish uses:

• sin lejlighed = her own apartment

If you said hendes lejlighed, it would usually mean another woman’s apartment, not the subject’s own.

Compare:

• Hun bor i sin lejlighed. = She lives in her own apartment.
• Hun bor i hendes lejlighed. = She lives in her apartment, but her refers to someone else.

This is one of the most important differences from English, since English uses her for both.

Because sin/sit/sine must agree with the noun that follows.

The noun here is:

• lejlighed = a common-gender noun (en lejlighed)

So the correct form is:

• sin lejlighed

The forms are:

• sin for en-words
• sit for et-words
• sine for plural nouns

Examples:

• sin bil
• sit hus
• sine bøger

Because i is the normal preposition for being in an apartment, room, house, city, etc.

So:

• i lejligheden = in the apartment
• i huset = in the house
• i byen = in the city

Using i here is very natural Danish.

In this sentence, lejlighed means apartment or flat.

But the word can also mean opportunity or occasion in other contexts. Danish has several words like this that depend on context.

Examples:

• en lejlighed i København = an apartment in Copenhagen
• jeg fik lejlighed til at tale med hende = I got the opportunity to speak with her

So in this sentence, the meaning is clearly the housing-related one.

Yes, very natural.

It sounds like standard everyday Danish and could easily be used in conversation or writing. The structure is common:

• subject: Hun
• modal phrase: vil gerne have
• object: et mere moderne køkken
• prepositional phrase: i sin lejlighed

A native speaker would understand it immediately, and it does not sound awkward or overly formal.