Jeg vil give de gamle bøger væk.

Because vil is a modal verb, and in Danish modal verbs are followed by the bare infinitive.

So Danish says:

• Jeg vil give ...
• not Jeg vil at give ...

This is like English I will give, not I will to give.

Other common Danish modal verbs work the same way:

• Jeg kan læse = I can read
• Jeg skal gå = I must / will go
• Jeg må vente = I may / must wait

Vil can mean several closely related things, depending on context:

• want to
• intend to
• sometimes simply will

So Jeg vil give de gamle bøger væk can sound like:

• I want to give the old books away
• I’m going to give the old books away
• I intend to give the old books away

A useful thing for English speakers to know is that Danish often uses the present tense for plain future meaning, so vil often carries a sense of willingness, desire, or intention, not just neutral future.

Yes. Give væk is a fixed verb expression, often called a particle verb.

Here:

• give = give
• væk = away

Together, give væk means give away.

This works much like English phrasal verbs:

• smide væk = throw away
• dele ud = hand out
• skrive ned = write down

So you should learn give væk as a unit of meaning, even though the two parts can be separated in the sentence.

Because in Danish particle verbs, the particle often comes after the object.

So the normal order here is:

• give + object + væk

That gives:

• Jeg vil give de gamle bøger væk.

This is very similar to English give the old books away.

In Danish, putting væk directly after give here would usually sound unnatural:

• natural: give de gamle bøger væk
• unnatural or much less standard here: give væk de gamle bøger

So for this kind of sentence, it is safest to keep the order:

verb + object + particle

Because Danish normally does not use the noun’s definite ending when there is a plural definite article plus an adjective before the noun.

So:

• bøger = books
• bøgerne = the books

But when you add an adjective, Danish uses:

• de gamle bøger = the old books

not:

• de gamle bøgerne

This is an important pattern:

• bilen = the car

• den gamle bil = the old car

• husene = the houses
• de store huse = the big houses

So after den/det/de + adjective, the noun usually appears without the definite ending.

Because adjectives in Danish change form depending on the noun phrase.

The basic form is:

• gammel = old

But with:

• plural nouns
• or definite noun phrases with den/det/de

the adjective usually takes -e:

• en gammel bog = an old book
• gamle bøger = old books
• de gamle bøger = the old books

So in your sentence, gamle is used because the noun phrase is plural and definite.

Here, de is a definite article, meaning the for plural nouns:

• de gamle bøger = the old books

Danish de can also be a pronoun meaning they, so the spelling is the same, but the function is different.

Compare:

• De kommer i morgen. = They are coming tomorrow.
  Here De is a pronoun.

• de gamle bøger = the old books
  Here de is an article.

So in your sentence, de does not mean they.

This is a very common question.

As pronouns:

• de = they (subject form)
• dem = them (object form)

Examples:

• De kommer. = They are coming.
• Jeg ser dem. = I see them.

But in your sentence, de is not a pronoun at all. It is the plural definite article meaning the:

• de gamle bøger = the old books

So do not replace it with dem here.

Yes, but the meaning changes.

• de gamle bøger = the old books / specific old books
• gamle bøger = old books / old books in general, or some old books

So:

• Jeg vil give de gamle bøger væk
  means you are talking about a particular set of old books.

• Jeg vil give gamle bøger væk
  sounds more general, like you want to give away old books as a category, or some old books not specifically identified.

So the presence of de makes the noun phrase definite.

Usually, no—not in normal neutral Danish.

The standard order is:

• Jeg vil give de gamle bøger væk.

With this type of verb, Danish normally places the object before the particle:

• tage jakken af = take the jacket off
• skrive det ned = write it down
• give bøgerne væk = give the books away

So for learners, the safest pattern is:

verb + object + particle

That is why Jeg vil give de gamle bøger væk is the natural version.