De vil levere de her stole i morgen, men den der lampe kommer først næste uge.

Because it is the first word of the sentence. Danish capitalizes the first word of a sentence just like English does.

So here, the capital D does not tell you anything special by itself.

In this sentence, it means they.

Danish does have a very formal De meaning you, but that is mostly found in especially formal or old-fashioned writing. Since De is at the start of the sentence, capitalization alone cannot tell you which one it is. You have to use context.

Yes, vil often corresponds to English will.

Here, vil levere means something like will deliver. In Danish, vil can express:

• future meaning
• intention
• willingness

In a sentence like this, it is most naturally understood as a future event: They will deliver...

Because Danish often uses the present tense for future events, especially when the event is planned, expected, or scheduled.

So:

• lampen kommer i morgen = the lamp is coming tomorrow
• lampen kommer først næste uge = the lamp won’t arrive until next week

This is very similar to English phrases like The train leaves tomorrow.

Yes, de her is a very common everyday way to say these.

So:

• de her stole = these chairs

You may also meet the more formal/written word disse:

• disse stole = these chairs

In ordinary spoken Danish, de her is extremely common.

They are both demonstrative expressions, but they differ in number and distance:

• de her = these (plural, near the speaker)
• den der = that (singular common gender, farther away)

So the sentence contrasts:

• de her stole = these chairs
• den der lampe = that lamp

The words her and der are literally here and there, but in these expressions they function like part of this/these/that.

Because Danish demonstratives must agree with the noun in number and, in the singular, gender.

• de is used for plural
• den is used for singular common gender
• det is used for singular neuter

So:

• de her stole = plural
• den der lampe = singular common gender

If the noun were neuter, you would get something like:

• det der bord = that table

Because when Danish uses a demonstrative like de her or den der, the demonstrative itself already makes the noun definite.

So Danish says:

• de her stole
• den der lampe

not:

• de her stolene
• den der lampen

This is an important pattern: after these demonstratives, the noun is normally in its basic singular or plural form, not the suffixed definite form.

Because Danish plural formation is not completely regular. Different nouns form the plural in different ways.

For stol:

• singular: stol
• plural: stole

This is a very common plural pattern in Danish, but it is something you often just have to learn with each noun.

Here, først does not mean first in the sense of before everything else. In time expressions like this, it often means:

• not until
• only

So:

• den der lampe kommer først næste uge

means:

• that lamp won’t arrive until next week
• or more literally, that lamp only comes next week

This is a very common use of først in Danish.

Because Danish time expressions do not all behave the same way.

With words like næste, sidste, and denne, Danish often uses the time phrase without a preposition:

• næste uge = next week
• sidste år = last year
• denne måned = this month

But i morgen is a fixed expression meaning tomorrow.

So both are normal:

• i morgen
• næste uge

Because men joins two main clauses, and after men Danish keeps normal main-clause word order unless something else is placed first.

Here the second clause starts with the subject:

• den der lampe = subject
• kommer = verb

So the order is simply:

• den der lampe kommer

Danish only puts the verb before the subject in a main clause when some other element comes first:

• I morgen kommer de = Tomorrow they are coming

But after men, there is no automatic inversion. The clause just starts normally with the subject.