I det lille rum ligger der allerede en madras på gulvet.

That part is a prepositional phrase meaning in the small room.

• i = in
• det lille rum = the small room

Danish often moves a place phrase to the front of the sentence to set the scene or emphasize location. English can do this too:

• In the small room, there is already a mattress on the floor.

When something other than the subject comes first in a Danish main clause, the word order changes because of the verb-second rule.

Because rum is a neuter noun in Danish.

Danish nouns usually have one of two genders:

• common gender → takes en
• neuter → takes et

Since the noun is et rum (a room), the definite form in this structure is det:

• et rum = a room
• det rum = that/the room
• det lille rum = the small room

If the noun were common gender, you would use den instead.

This is a very common question. Even though rum is neuter, the adjective is lille, not lillet, because the noun phrase is definite: det lille rum.

Compare:

• et lille rum = a small room
• det lille rum = the small room

In indefinite neuter singular, many adjectives take -t:

• et stort rum = a big room

But in the definite form, adjectives usually take -e:

• det store rum = the big room

With lille, the standard form is just lille in both places:

• et lille rum
• det lille rum

So det lille rum is completely normal.

Because Danish main clauses normally follow the V2 rule: the finite verb must be in the second position.

Here, the first position is taken by I det lille rum. So the verb has to come next:

• I det lille rum | ligger | der allerede en madras på gulvet

That is why you get ligger der rather than der ligger.

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

• Der ligger allerede en madras på gulvet.

Both are grammatical, but the version with I det lille rum first puts extra focus on the location.

Yes, it is very similar to the existential there in English, as in:

• There is a mattress on the floor.

In this sentence, der does not mean a physical place like over there. It is a grammatical filler used in existence/presence statements.

So:

• der ligger en madras ≈ there is a mattress lying / there is a mattress

It helps introduce something that exists in a place.

Danish very often uses position verbs where English would simply use be.

Here, ligger literally means lies / is lying, and it is used because a mattress is in a horizontal resting position.

Common Danish position verbs include:

• ligge = lie
• stå = stand
• sidde = sit

So Danish prefers:

• Der ligger en madras på gulvet.

where English would usually say:

• There is a mattress on the floor.

Using er is not impossible in all contexts, but ligger sounds more natural here because it describes how the object is positioned.

Because madras is a common gender noun.

Its basic form is:

• en madras = a mattress

So in the sentence:

• en madras = a mattress

You just have to learn the gender with each noun. A useful habit is to memorize nouns together with their article:

• et rum
• en madras

Because the sentence is introducing the mattress as something not previously specified.

• en madras = a mattress
• madrassen = the mattress

If you say en madras, you are presenting it as one mattress in the room. If you said madrassen, it would sound like the listener already knows which mattress you mean.

So this sentence is naturally phrased with the indefinite noun:

• der ligger allerede en madras ...

Because the floor is understood as a specific, known floor: the floor of that room.

So Danish uses the definite form:

• gulv = floor
• gulvet = the floor

This is similar to English, where we also usually say:

• on the floor

not

• on a floor

In a room, the floor is normally treated as an obvious, identifiable thing.

Allerede means already.

It tells you that the mattress is there earlier than expected, or that the situation is already true.

So:

• der ligger allerede en madras på gulvet
  means
• there is already a mattress on the floor

Its placement here is natural in Danish, between der and en madras.

Yes, that is possible.

Compare:

• I det lille rum ligger der allerede en madras på gulvet.
• Der ligger allerede en madras på gulvet i det lille rum.

The difference is mostly about focus.

The original sentence puts the location first, so it feels like:

• As for the small room, there is already a mattress on the floor there.

The version starting with Der ligger ... is a more neutral existence statement, and the location comes later.

Not exactly a fixed expression, but it is a very common pattern in Danish.

You often get:

• Der ligger ...
• Der står ...
• Der sidder ...

And when another element is moved to the front, the verb comes before der because of V2:

• På bordet står der en kop.
• I haven sidder der en fugl.
• I det lille rum ligger der allerede en madras på gulvet.

So the pattern is very productive and worth learning.

Normally, not in this kind of existential sentence.

Danish typically needs der when introducing that something exists somewhere:

• Der ligger en madras på gulvet.

If you remove der, the sentence would usually sound wrong or incomplete in this structure:

• I det lille rum ligger allerede en madras på gulvet
  This is not the normal way to express the idea.

So for learners, it is best to keep der in existence statements like this.

A helpful breakdown is:

• I det lille rum = fronted place phrase
• ligger = finite verb
• der = existential filler
• allerede = adverb
• en madras = the thing that exists
• på gulvet = more location information

So the structure is roughly:

[place] + [verb] + der + [adverb] + [noun phrase] + [place]

A very literal version would be:

• In the small room lies there already a mattress on the floor.

That sounds odd in English, but it reflects the Danish grammar quite well.