Det er rengøringen, der tager lang tid, fordi der hurtigt kommer støv i trappeopgangen.

This is a very common Danish emphasis pattern called a cleft sentence:

Det er rengøringen, der tager lang tid ...

It means something like It’s the cleaning that takes a long time ...

The speaker is highlighting rengøringen as the important part.
A more neutral version would be:

Rengøringen tager lang tid, fordi der hurtigt kommer støv i trappeopgangen.

So the Det er ... der ... structure adds focus, not a big change in basic meaning.

Because rengøringen is the definite form: the cleaning.

In Danish, definiteness is often added as an ending:

• en rengøring = a cleaning
• rengøringen = the cleaning

In this sentence, the speaker is referring to a specific activity or task already understood in context, so the definite form is natural.

Here der introduces the clause that describes rengøringen:

... der tager lang tid = ... that takes a long time

In English, we might say that or sometimes leave it out, but Danish often uses der when the word being referred to is the subject of the following clause.

So:

• rengøringen tager lang tid = the cleaning takes a long time
• det er rengøringen, der tager lang tid = it’s the cleaning that takes a long time

This der is not the same as the later der in fordi der hurtigt kommer støv.

In Danish, tage lang tid is the normal expression for to take a long time.

So:

• Det tager lang tid = It takes a long time
• Rengøringen tager lang tid = The cleaning takes a long time

Using er lang tid would not sound natural here. Danish uses tage in the same way English uses take.

This der means something like English there in sentences such as:

• There is dust
• There comes dust
• more naturally, Dust quickly appears / accumulates

So this is a different der from the earlier relative der.

Here it is part of an existential construction, where Danish says that something comes, is, or exists somewhere:

• der er støv = there is dust
• der kommer støv = dust appears / dust comes / dust builds up

So the two der words in the sentence have different jobs.

Literally, yes, it says dust comes, but idiomatically it means:

• dust appears
• dust collects
• dust builds up

Danish often uses komme in ways that sound broader than English come. In this sentence, der hurtigt kommer støv means that dust quickly starts to appear or accumulate in the stairwell.

In this sentence, hurtigt modifies the event: dust appears quickly.

The clause is:

fordi der hurtigt kommer støv i trappeopgangen

This word order is natural in Danish. The adverb is placed early in the clause, before the verb kommer.

You may also hear slightly different placements in other contexts, but this version is very standard and natural here. The important point for a learner is that Danish adverbs do not always go in exactly the same place as in English.

Trappeopgang means stairwell, staircase area, or apartment building entrance/stair area, depending on context.

• en trappeopgang = a stairwell / stair entrance
• trappeopgangen = the stairwell

The preposition i means in here, because the dust is appearing in that area.

So:

i trappeopgangen = in the stairwell

This is more specific than just på trappen (on the stairs). A trappeopgang is usually the whole shared stair area, not only the steps themselves.

For the same reason as rengøringen: Danish uses the definite ending when referring to a specific, known place.

• en trappeopgang = a stairwell
• trappeopgangen = the stairwell

In context, the speaker means a particular stairwell, probably the one in the building being discussed.

The commas separate the main parts of the sentence:

Det er rengøringen, der tager lang tid, fordi der hurtigt kommer støv i trappeopgangen.

The first comma marks the start of the der clause connected to the cleft structure.
The second comma separates the main statement from the fordi clause.

Comma rules in Danish are not always identical to English comma rules, but commas before clauses like fordi are very common in written Danish.

A plain, neutral version would be:

Rengøringen tager lang tid, fordi der hurtigt kommer støv i trappeopgangen.

That means the same basic thing, but without the special emphasis on rengøringen.

So the difference is roughly:

• Det er rengøringen, der tager lang tid ... = It’s the cleaning that takes a long time ...
• Rengøringen tager lang tid ... = The cleaning takes a long time ...

The first one sounds a bit more contrastive or explanatory.

Yes, støv is a noun, but here it works like an uncountable noun, similar to English dust.

So Danish normally says:

• der er støv = there is dust
• der kommer støv = dust appears / dust builds up

You would not normally use an article here, just as English normally says dust, not a dust.

In some relative clauses, Danish can use som, but in this cleft pattern Det er ... der ..., der is the normal choice:

Det er rengøringen, der tager lang tid.

So for a learner, the safest rule is:

• after Det er X ... in this kind of sentence, use der
• especially when the following clause has X as its subject

That will sound natural and correct.