Jeg vil smide de gamle aviser ud, før gæsterne kommer.

Because vil is a modal verb. After Danish modal verbs such as vil, kan, skal, må, bør, and tør, the next verb appears in the bare infinitive, without at.

So:

• Jeg vil smide ...
• not Jeg vil at smide ...

This is similar to English I will throw..., not I will to throw....

It can mean either, depending on context and tone. Vil often expresses:

• desire: want to
• willingness: am willing to
• intention/future: will / am going to

So Jeg vil smide de gamle aviser ud can sound like either I want to throw out the old newspapers or I’ll throw out the old newspapers. Danish uses vil in a wider way than English will.

Because smide ud is a particle verb, much like English throw out. In Danish, the object often goes between the main verb and the particle:

• smide aviserne ud
• smide dem ud

So in your sentence, de gamle aviser sits between smide and ud.

This is completely normal Danish word order.

When a noun is definite and also has an adjective in front of it, Danish normally uses:

• den/det/de
  • adjective + noun

So:

• aviserne = the newspapers
• de gamle aviser = the old newspapers

The noun stays in its basic plural form aviser, not aviserne, because the definiteness is shown by de.

Because adjectives in Danish change form.

The basic pattern is:

• en gammel avis = an old newspaper
• et gammelt hus = an old house
• gamle aviser = old newspapers
• de gamle aviser = the old newspapers

In plural, and also in definite adjective phrases, the adjective usually takes -e. That is why you get gamle here.

This is a very common thing learners notice.

• gæsterne = the guests
• de gamle aviser = the old newspapers

Why the difference?

Because gæsterne has no adjective, so Danish marks definiteness with the noun ending -ne.

But de gamle aviser does have an adjective, so Danish uses de before the adjective instead of putting the definite ending on the noun.

So:

• without adjective: aviserne, gæsterne
• with adjective: de gamle aviser, de venlige gæster

Because Danish often uses the present tense for future events, especially in time clauses introduced by words like før, når, and efter.

So før gæsterne kommer is the normal way to say before the guests arrive.

English actually does something similar:

• before the guests arrive
• not usually before the guests will arrive

So the Danish present tense here is completely natural.

Because før gæsterne kommer is a subordinate clause.

In Danish:

• main clauses usually follow verb-second order
• subordinate clauses usually have subject before verb

Compare:

• Gæsterne kommer. = main clause
• før gæsterne kommer = subordinate clause

So after før, the subject gæsterne comes before the verb kommer.

Because før gæsterne kommer is a subordinate clause, and many Danish writers put a comma before such clauses.

This sentence uses what is often called start comma style. In modern Danish, you may also see the sentence written without that comma:

• Jeg vil smide de gamle aviser ud før gæsterne kommer.

So the comma is common and correct in many contexts, but you may encounter both versions.

In de gamle aviser, de is the definite plural article, meaning the.

• de gamle aviser = the old newspapers

Dem is usually the object pronoun meaning them:

• Jeg smider dem ud. = I’m throwing them out.

So in your sentence, de is correct because it is an article, not a pronoun.