Hun vil flytte ind i den nye lejlighed på fredag.

In Danish, vil can cover both ideas:

• future/intention: will / is going to
• wanting/willingness: wants to

In this sentence, it most naturally sounds like future intention: she plans or expects to move in on Friday.

So a good way to think of it is:

• Hun vil flytte ind ... = she will / is going to / intends to move in ...

Context usually tells you whether vil is more about the future or about desire.

Because vil is a modal verb, and after a modal verb Danish uses the bare infinitive.

So you say:

• hun vil flytte
• hun kan komme
• hun skal arbejde

Not:

• hun vil at flytte

This is similar to English she will move, not she will to move.

It is best understood as a verb + particle expression:

• flytte = move
• flytte ind = move in

So ind is an important part of the meaning here.
Without ind, flytte just means move in a broader sense.

Compare:

• Hun flytter = She is moving
• Hun flytter ind = She is moving in

Because they do different jobs:

• ind gives the idea of movement inward
• i introduces the place/object being entered

So:

• flytte ind = move in
• flytte ind i lejligheden = move into the apartment

You can say Hun flytter ind if you do not say where.
But if you mention the place, Danish normally uses i:

• flytte ind i den nye lejlighed

Both can mean the apartment, but they are used differently.

• lejligheden = the apartment
• den nye lejlighed = the new apartment

When there is an adjective before the noun in a definite noun phrase, Danish usually uses this pattern:

• den/det/de + adjective + noun

So:

• lejligheden = the apartment
• den nye lejlighed = the new apartment

You do not normally say den nye lejligheden.

Because lejlighed is a common gender noun, an en-word:

• en lejlighed = an apartment

Common gender nouns take den in the definite phrase:

• den nye lejlighed

If it were a neuter noun, a et-word, you would use det instead.

Because the noun phrase is definite.

In Danish, adjectives usually take -e in definite forms:

• en ny lejlighed = a new apartment
• den nye lejlighed = the new apartment

So the pattern is:

• indefinite singular: ny
• definite: nye

That is why you get den nye lejlighed.

It usually means this coming Friday or on Friday, depending on context.

Danish commonly uses på with days when talking about a specific day:

• på mandag = on Monday
• på fredag = on Friday

So på fredag is very natural here.
If the context is clear, it normally refers to the Friday nearest in the future.

Yes. This is a normal Danish main clause:

• Hun = subject
• vil = finite verb
• flytte ind = infinitive verb phrase
• i den nye lejlighed = place/direction phrase
• på fredag = time phrase

A key rule in Danish main clauses is V2: the finite verb comes in the second position.

That is why vil comes right after Hun.

Yes, absolutely:

• På fredag vil hun flytte ind i den nye lejlighed.

That is very natural Danish.

Notice what happens: when På fredag moves to the front, the finite verb vil still stays in second position, so hun moves after it.

This is a very important Danish word-order pattern.

You invert the subject and the finite verb:

• Vil hun flytte ind i den nye lejlighed på fredag?

So:

• statement: Hun vil flytte ind ...
• question: Vil hun flytte ind ... ?

That verb-subject inversion is standard in Danish yes/no questions.

Usually flytte, nye, and lejlighed cause the most trouble.

A few things to watch out for:

• y in flytte and nye is a Danish vowel that English does not really have
• lejlighed has a soft d at the end, not a strong English d
• Danish unstressed syllables are often reduced, so the rhythm may sound softer than an English speaker expects

A rough learner-friendly guide might be:

• Hun ≈ hoon
• vil ≈ vil
• flytte ≈ FLÜ-də
• ind i ≈ in ee
• den nye lejlighed ≈ den NY-uh LYE-lee-hedh
• på fredag ≈ po FREY-dah

But that is only approximate. For pronunciation, it is best to listen to native audio if possible.