Hendes lejlighed er ikke den mest moderne i bygningen, men køkkenet er lyst og hyggeligt.

Because Danish normally uses a possessive directly before the noun, without an article.

• hendes lejlighed = her apartment
• not den hendes lejlighed

This is similar to English, where we say her apartment, not the her apartment.

Also notice that the noun stays in its basic singular form after a possessive:

• hendes lejlighed
• min bil
• vores hus

So the possessive already makes the noun definite enough by itself.

Because sin/sit/sine is a reflexive possessive, and it refers back to the subject of the clause.

In this sentence, the grammatical subject is lejlighed:

• Hendes lejlighed er ikke ...

So her is not the subject here; the apartment is. That means Danish uses hendes, not sin.

Compare:

• Hun elsker sin lejlighed = She loves her own apartment
  Here hun is the subject, so sin can refer back to hun.

• Hendes lejlighed er stor = Her apartment is big
  Here the subject is lejlighed, so sin would not work.

Because in a normal Danish main clause, the finite verb usually comes before sentence adverbs like ikke.

So the pattern here is:

• Hendes lejlighed = subject
• er = finite verb
• ikke = negation

That gives:

• Hendes lejlighed er ikke ...

This is a very common Danish word-order pattern:

• Han kommer ikke i dag
• Vi har ikke tid
• Det er ikke svært

So even though English says is not, Danish often places the negation after the verb in this kind of sentence.

Because Danish usually uses den/det/de with a definite superlative when talking about a specific item.

Here den mest moderne means:

• the most modern one

It refers back to lejlighed, which is a common gender noun, so the sentence uses den.

You can think of it like this:

• Hendes lejlighed er ikke den mest moderne i bygningen
• literally: Her apartment is not the most modern in the building
• more fully: ... not the most modern apartment in the building

The noun lejlighed is understood, so it does not need to be repeated.

Because moderne is one of those adjectives whose basic form already ends in -e, and it does not add an extra ending here.

Danish adjectives often change form, but moderne stays the same in many contexts:

• en moderne lejlighed
• et moderne køkken
• den moderne stil
• den mest moderne

So in this sentence, moderne is simply the correct form.

i bygningen literally means in the building.

The noun is:

• en bygning = a building

Its definite singular form is made by adding -en:

• bygningen = the building

So:

• i bygningen = in the building

This phrase means that the comparison is being made among the apartments in that building.

Because they describe køkkenet, and køkken is a neuter noun:

• et køkken = a kitchen
• køkkenet = the kitchen

When an adjective is used predicatively after er and describes a singular neuter noun, it often takes -t:

• køkkenet er lyst
• køkkenet er hyggeligt

Compare with common gender:

• en stue er lys
• en stue er hyggelig

So the -t ending shows agreement with the neuter noun køkkenet.

Because men is a coordinating conjunction, and after it Danish starts a new main clause with normal main-clause word order.

So the second clause is simply:

• køkkenet = subject
• er = verb
• lyst og hyggeligt = complement

That gives:

• men køkkenet er lyst og hyggeligt

The conjunction men does not force inversion by itself.

Compare:

• Hun er træt, men hun arbejder videre
• Det er gammelt, men det fungerer

So the word order stays straightforward: subject + verb.

Yes. Danish often leaves out a noun when it is already understood from the context.

Here, den mest moderne really means:

• the most modern apartment
• or more literally, the most modern one

Since lejlighed has already been mentioned, Danish does not need to repeat it.

So these are both possible in principle:

• Hendes lejlighed er ikke den mest moderne i bygningen
• Hendes lejlighed er ikke den mest moderne lejlighed i bygningen

The first version sounds more natural because it avoids unnecessary repetition.