Efter rengøringen ligger der stadig støv på hylden.

Because rengøringen is the definite form: the cleaning.

The base noun is en rengøring = a cleaning.
In Danish, definiteness is often added as a suffix:

• en rengøring = a cleaning
• rengøringen = the cleaning

So efter rengøringen means after the cleaning, referring to a specific cleaning event.

Compare:

• efter rengøringen = after the cleaning
• efter en rengøring = after a cleaning
• efter rengøring = after cleaning / after doing cleaning in general

This is because Danish main clauses follow the V2 rule: the finite verb normally comes in second position.

Here, Efter rengøringen has been placed first as a time expression. Once that happens, the finite verb must come next:

• Efter rengøringen
  • ligger
    • der stadig støv på hylden.

A more neutral order would be:

• Der ligger stadig støv på hylden efter rengøringen.

So ligger is not randomly moved; it is in second position because Efter rengøringen is occupying the first slot.

Here der is a dummy subject or formal subject, similar to English there in sentences like There is still dust on the shelf.

It does not mean a physical place here.

So in this sentence:

• der = existential/presentational there
• not there as in over there

Danish often uses der in sentences that introduce the existence or presence of something:

• Der er en bog på bordet. = There is a book on the table.
• Der ligger støv på hylden. = There is dust lying on the shelf.

Danish often prefers a position verb where English would simply use be.

Here, ligger comes from at ligge = to lie. It is used because the dust is understood as resting on a surface.

So Danish says, more literally:

• there lies still dust on the shelf

This sounds natural in Danish.

You could also hear:

• Der er stadig støv på hylden.

That is also correct, but ligger is a bit more concrete and physical. Danish commonly uses verbs like:

• ligge for things lying/resting
• stå for things standing
• sidde for things sitting/attached in certain positions

Because støv is usually an uncountable mass noun, like English dust.

Just as English normally says there is dust, not there is a dust, Danish says:

• der ligger støv
• not der ligger et støv

If you want something countable, Danish would use another noun, for example:

• et støvkorn = a speck/grain of dust
• støvpartikler = dust particles

So støv here works like a mass noun and does not need an article.

Grammatically, it behaves like a singular mass noun.

That is why the sentence uses the singular present form ligger. Danish does not normally use a plural form when talking about dust in a general sense.

So even though dust consists of many tiny particles, the language treats støv as one uncountable substance, much like:

• vand = water
• sand = sand
• ris = rice (often mass-like in use)

Stadig is an adverb meaning still, and its position follows normal Danish word-order patterns.

In this clause, the structure is roughly:

• Efter rengøringen / ligger / der / stadig / støv / på hylden

Since der is the formal subject, stadig naturally comes after it.

You can also see the same thing in the more neutral version:

• Der ligger stadig støv på hylden.

So the placement of stadig is not arbitrary; it fits standard Danish sentence structure.

Because the dust is on the surface of the shelf, not inside it.

• på = on
• i = in

A shelf is usually thought of as a flat surface you place things on, so:

• på hylden = on the shelf

If something were inside a cabinet, drawer, or box, then i would be more natural.

Because hylden is the definite form: the shelf.

The noun is:

• en hylde = a shelf
• hylden = the shelf

So på hylden means on the shelf, probably a specific shelf already known from the context.

If the shelf were not specific, Danish could say:

• på en hylde = on a shelf

Yes. That is also a correct and natural sentence.

The difference is mostly about focus:

• Efter rengøringen ligger der stadig støv på hylden.
  This emphasizes the time frame first: after the cleaning...

• Der ligger stadig støv på hylden efter rengøringen.
  This starts more neutrally with the existence of the dust.

Both are good Danish. The original sentence fronts Efter rengøringen to highlight the contrast: even after cleaning, the dust is still there.