Nach dem Einzug ist die Wohnung endlich gemütlich.

Because nach takes the dative case when it means after in a time expression.

• der Einzug = nominative
• den Einzug = accusative
• dem Einzug = dative

So:

• nach dem Einzug = after the move-in

Also, Einzug is a masculine noun, so its dative singular article is dem.

Einzug is a noun meaning moving in or move-in.

It comes from the verb einziehen, which can mean to move into a new place.

In this sentence, nach dem Einzug means something like:

• after moving in
• after the move-in

It refers to the point when people have moved into the apartment and things are now settled.

In German, all nouns are capitalized.

So:

• der Einzug = the move-in
• die Wohnung = the apartment
• dem Einzug = to/after the move-in

That is why Einzug and Wohnung begin with capital letters.

German often puts time expressions at the beginning of the sentence.

So Nach dem Einzug is placed first to set the time frame:

• After moving in, the apartment is finally cozy.

When something other than the subject comes first, German still keeps the finite verb in second position. That is why the sentence continues with:

• Nach dem Einzug ist ...

not

• Nach dem Einzug die Wohnung ist ...

This is because German is a verb-second language.

The first position is occupied by Nach dem Einzug, so the finite verb must come next:

• Nach dem Einzug
  • ist
    • die Wohnung
      • ...

Even though die Wohnung comes after the verb, it is still the subject of the sentence.

A more neutral order would be:

• Die Wohnung ist nach dem Einzug endlich gemütlich.

Both are correct, but the original version emphasizes the time expression more.

Because die Wohnung is in the nominative case and it agrees with the verb ist.

Here:

• die Wohnung = the apartment
• ist = is

The apartment is what is cozy, so it is the subject.

German does not require the subject to come first in the sentence. Word order is more flexible than in English.

Because gemütlich is used here as a predicate adjective, after the verb sein.

Predicate adjectives in German do not take adjective endings:

• Die Wohnung ist gemütlich. = The apartment is cozy.

But when the adjective comes before a noun, it does take an ending:

• eine gemütliche Wohnung = a cozy apartment

So:

• die Wohnung ist gemütlich ✅
• die Wohnung ist gemütliche ❌

Endlich means finally or at last.

It adds the idea that this cozy state was expected or delayed. In other words, the apartment was probably not cozy before, but now it finally is.

In this sentence, it fits naturally before the adjective:

• ist die Wohnung endlich gemütlich

You could also say:

• Die Wohnung ist endlich gemütlich.

That is probably the most straightforward word order if the time phrase is not being emphasized.

Yes, but there is a nuance.

• der Einzug focuses on moving into a specific place
• der Umzug means the move / relocation more generally

So:

• nach dem Einzug = after moving into the apartment
• nach dem Umzug = after the whole move

In this sentence, Einzug is very natural because the sentence is about how the apartment feels once people are living in it.

Because the sentence describes the apartment’s current state:

• the apartment is finally cozy

The phrase nach dem Einzug gives the time background, but the main statement is about how things are now.

German often uses the present tense in exactly this way, just as English does.

Yes. German allows several word orders, with small changes in emphasis.

For example:

• Nach dem Einzug ist die Wohnung endlich gemütlich.
  • Emphasizes after moving in
• Die Wohnung ist nach dem Einzug endlich gemütlich.
  • More neutral, starts with the subject
• Endlich ist die Wohnung nach dem Einzug gemütlich.
  • Stronger emphasis on finally

The most important rule is that the finite verb stays in second position in a main clause.