Jeg lader børnene vælge, hvilke bøger vi skal beholde, og hvilke vi skal give væk.

Lader is the present tense of at lade, which often means to let or to allow.

So Jeg lader børnene vælge means I let the children choose.

In this sentence, lade is being used as a causative/permission verb:

• Jeg lader dem gøre det = I let them do it
• Hun lader mig tale færdig = She lets me finish speaking

It does not mean load here, even though that may be the first meaning some learners meet in dictionaries.

After lade, Danish normally uses a bare infinitive, just like English after let.

So:

• Jeg lader børnene vælge not
• Jeg lader børnene at vælge

This is parallel to English:

• I let the children choose not
• I let the children to choose

Other verbs can behave differently, but lade takes the infinitive without at.

Because barn is an irregular noun.

Its forms are:

• et barn = a child
• barnet = the child
• børn = children
• børnene = the children

So børnene is the definite plural form: the children.

This is one of those noun patterns you mostly just have to learn as a set.

Because hvilke bøger means which books, and after hvilke you normally use the noun in the indefinite plural form.

So:

• hvilke bøger = which books not
• hvilke de bøger and normally not
• hvilke bøgerne

Compare:

• Hvilken bog vil du have? = Which book do you want?
• Hvilke bøger vil du have? = Which books do you want?

The word hvilke already makes the noun specific enough in this kind of structure.

Hvilke means which and introduces an embedded question.

So:

• hvilke bøger vi skal beholde = which books we should keep
• hvilke vi skal give væk = which ones we should give away

It is not asking the listener directly. Instead, it is part of what the children are choosing.

So the structure is roughly:

• I let the children choose [which books we should keep] and [which ones we should give away].

Because the sentence is contrasting two groups:

• the books we should keep
• the books we should give away

Repeating hvilke makes the structure clear and balanced:

• hvilke bøger vi skal beholde, og hvilke vi skal give væk

In English, we often do the same:

• which books we should keep, and which we should give away

The second hvilke stands for which ones.

Because this is an embedded question, not a main question.

In a main question, Danish often has inversion:

• Hvilke bøger skal vi beholde? = Which books should we keep?

But inside a larger sentence, the word order becomes more like a statement:

• ... hvilke bøger vi skal beholde = ... which books we should keep

So:

• main question: skal vi
• embedded question: vi skal

This is very common in Danish and is similar to English:

• direct question: Which books should we keep?
• embedded question: ... which books we should keep

Here skal most naturally means something like should, are to, or are supposed to.

So:

• hvilke bøger vi skal beholde = which books we should keep
• hvilke vi skal give væk = which ones we should give away

It is not just a neutral future marker here. It suggests a decision, plan, or obligation.

Depending on context, skal can mean:

• obligation: must / should
• planned future: will / are going to
• instruction or arrangement: are to

In this sentence, the idea is that the children are deciding what belongs in each category.

Beholde means to keep, to retain, or to hold on to something.

It is a separate verb from holde.

Compare:

• holde = hold, keep, stop, last, depending on context
• beholde = keep, retain

Examples:

• Jeg vil beholde bogen. = I want to keep the book.
• Kan du holde tasken? = Can you hold the bag?

So in your sentence, beholde is exactly the right verb for keep.

Because give væk means give away, not just give.

Compare:

• give = give
• give væk = give away

Examples:

• Jeg giver hende en bog. = I give her a book.
• Jeg giver bogen væk. = I give the book away.

In your sentence, the meaning is that the books are being removed from your possession, so give væk is the natural choice.

Yes, it behaves a lot like a phrasal verb.

You will often see the parts separated in finite clauses:

• Vi giver bøgerne væk. = We give the books away.

And in infinitive structures, you also commonly see:

• at give væk
• skal give væk

So in your sentence:

• vi skal give væk

That is completely normal Danish word order.

Because bøger is understood from the first part.

The full idea is:

• hvilke bøger vi skal beholde, og hvilke bøger vi skal give væk

But Danish, like English, often leaves out a repeated noun when it is obvious:

• which books we should keep, and which we should give away

So the second hvilke effectively means which ones.

The commas help separate the main clause from the embedded clauses and make the parallel structure easier to read.

Main clause:

• Jeg lader børnene vælge

Then come the two embedded question parts:

• hvilke bøger vi skal beholde
• og hvilke vi skal give væk

So the commas show the sentence structure clearly.

You may also notice that Danish comma usage can vary somewhat depending on comma style taught or preferred, but in a sentence like this, the commas are there to mark the clause boundaries and improve readability.

Yes, that is a very close structural translation.

A literal breakdown is:

• Jeg = I
• lader = let
• børnene = the children
• vælge = choose
• hvilke bøger = which books
• vi skal beholde = we should keep / we are to keep
• og hvilke = and which ones
• vi skal give væk = we should give away

A more natural English translation might be:

• I let the children choose which books we should keep and which ones we should give away.

So yes, your more literal version is grammatically very close to the Danish structure.