I kælderen er der et vaskeri med to vaskemaskiner.

I kælderen means in the basement.

It comes first because Danish often puts a place or time expression at the beginning when that is the setting for the sentence. English can do something similar:

• In the basement, there is...
• There is ... in the basement.

Both are possible in Danish too, but starting with I kælderen makes the location the frame for the rest of the sentence.

Also, kælderen is the definite form of kælder:

• en kælder = a basement
• kælderen = the basement

So i kælderen literally means in the basement.

This is because of normal Danish word order, often called the V2 rule.

In main clauses, the finite verb usually comes in the second position. Since I kælderen has been placed first, the verb er must come next:

• I kælderen er der et vaskeri ...

If the sentence started with der, then you would get:

• Der er et vaskeri i kælderen.

Both are correct, but the structure changes depending on what comes first.

So:

• Der er ... = when der starts the sentence
• I kælderen er der ... = when the location starts the sentence

Yes, it is very similar to existential there in English.

In this sentence, der does not mean a physical place like over there. It is a grammatical dummy subject, used in existence statements:

• Der er et vaskeri = There is a laundry room / laundromat

This is just like English:

• There is a problem
• There are two chairs

So in I kælderen er der et vaskeri, der is part of the structure meaning there is.

Because vaskeri is a neuter noun in Danish.

Danish nouns have two grammatical genders:

• common gender: takes en
• neuter gender: takes et

So:

• et vaskeri = a laundry room / laundromat

Its definite form is:

• vaskeriet = the laundry room / the laundromat

Unfortunately, noun gender usually has to be learned with each noun. There is no fully reliable rule that tells you whether a noun takes en or et.

They are related, but they do not mean the same thing.

• et vaskeri = a laundry room, laundry facility, or laundromat
• en vaskemaskine = a washing machine

So in the sentence:

• et vaskeri is the place/facility
• to vaskemaskiner are the machines inside it

You can break vaskemaskine into parts:

• vaske- = washing
• maskine = machine

And vaskemaskiner is just the plural form:

• en vaskemaskine = one washing machine
• to vaskemaskiner = two washing machines

Because med is a very natural and compact way in Danish to describe what something contains or comes with.

• et vaskeri med to vaskemaskiner = a laundry room with two washing machines

This works much like English with.

You could say something longer, such as a version meaning which has two washing machines, but med is simpler and more idiomatic here.

So med is not especially tricky in this sentence: it just links the laundry room to what it has.

Because after a number, Danish normally uses the plural noun without an article.

So:

• en vaskemaskine = a washing machine
• to vaskemaskiner = two washing machines

This is similar to English:

• a washing machine
• two washing machines

You do not say an extra some or the unless the meaning requires it.

The plural ending here is -r:

• vaskemaskine
• vaskemaskiner

Yes, absolutely.

Both are natural, but they emphasize things slightly differently:

• Der er et vaskeri i kælderen.
  More neutral: There is a laundry room in the basement.

• I kælderen er der et vaskeri.
  Starts by setting the location: In the basement, there is a laundry room.

So the difference is mainly about information structure and emphasis, not basic meaning.

Because they play different roles in the sentence.

• kælderen = the basement
  This refers to a specific basement, usually one already understood from context, such as the basement of a building.

• et vaskeri = a laundry room / laundry facility
  This introduces something as new information: there exists such a place in the basement.

This is very normal in both Danish and English:

• In the basement, there is a laundry room...

The location is already identifiable, but the thing being introduced is new.

It can depend on context.

Vaskeri can mean:

• a laundromat
• a laundry room / laundry facility in an apartment building, dorm, etc.

In this sentence, because it says in the basement and then mentions two washing machines, the most natural understanding is a shared laundry room in the building rather than a commercial laundromat.

So the word itself can cover more than one idea, and context tells you which one is meant.