Vi pakker først bøgerne ud, og derefter lægger hun sit sengetøj i skabet.

Because pakke ud is a separable verb, meaning to unpack.

In main clauses, Danish often splits this kind of verb:

• Vi pakker bøgerne ud = We unpack the books.

Here:

• pakker is the finite verb
• ud is the particle, and it usually comes later in the clause

This is similar to English phrasal verbs like pick up or take off, where the small word can be separated:

• We pick the books up

So pakker ... ud is completely normal Danish word order.

In Danish, the definite article is usually added to the end of the noun.

So:

• bog = book
• bøger = books
• bøgerne = the books

This is one of the biggest differences from English. Instead of saying the books, Danish often uses one word: bøgerne.

They both relate to sequence, but they are used a little differently:

• først = first
• derefter = after that / afterwards / then

In the sentence:

• Vi pakker først bøgerne ud = First, we unpack the books
• og derefter lægger hun sit sengetøj i skabet = and after that, she puts her bedding in the closet

So først introduces the first action, while derefter introduces the next one.

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

In the second clause, derefter has been moved to the front:

• derefter = first position
• lægger = second position
• hun = comes after the verb

So:

• Derefter lægger hun sit sengetøj i skabet

This is very typical Danish word order.
If there were no fronted adverb, you would say:

• Hun lægger sit sengetøj i skabet

Because Danish uses a reflexive possessive when the owner is the subject of the clause.

In this clause:

• hun = she
• she is also the one who owns the bedding

So Danish uses:

• sit = her own / his own / its own, agreeing with the noun

Compare:

• Hun lægger sit sengetøj i skabet = She puts her own bedding in the closet
• Hun lægger hendes sengetøj i skabet = She puts another woman’s bedding in the closet

So sit shows that the bedding belongs to hun.

Because sengetøj is a neuter singular noun, and the reflexive possessive has to agree with it.

The forms are:

• sin for common gender singular nouns
• sit for neuter singular nouns
• sine for plural nouns

Since:

• et sengetøj is neuter

you get:

• sit sengetøj

Grammatically, it is singular, but in meaning it often works like a collective noun such as bedding or bed linen in English.

So:

• et sengetøj = a set of bedding / bed linen
• sengetøj can refer to bedding in a general sense

That is why Danish uses sit and not a plural form here.

i means in, and skabet means the closet / the cabinet.

Breaking it down:

• et skab = a closet, cabinet, cupboard
• skabet = the closet / cabinet

So:

• i skabet = in the closet

The preposition i is used because the bedding is being placed inside something.

Yes, that is also possible.

Both of these are natural:

• Vi pakker først bøgerne ud
• Vi pakker bøgerne først ud

The difference is mostly one of emphasis and rhythm.

• først before the object can emphasize the sequence a bit more naturally in many contexts
• placing først later can sound like you are emphasizing when the unpacking happens

Both are understandable and grammatically fine.

Because the two clauses describe two separate actions, and the subject changes:

• Vi pakker først bøgerne ud = We unpack the books first
• derefter lægger hun sit sengetøj i skabet = after that, she puts her bedding in the closet

So the first clause has vi and the second has hun.
That is perfectly normal. The conjunction og simply links the two main clauses.

lægger is from lægge, which commonly means to lay, put, place.

In Danish, lægge is very often used where English would say put, especially when placing something somewhere:

• lægge noget i skabet = put something in the closet
• lægge bogen på bordet = put the book on the table

So lægger is exactly the normal verb to use here.