Til rengøringen bruger hun en blød klud, fordi der er meget støv på bordet.

Because Danish is a verb-second language in main clauses.

When Til rengøringen is placed first, it takes the first slot in the sentence, so the finite verb bruger must come next:

Til rengøringen bruger hun en blød klud.

If the subject comes first instead, then you get:

Hun bruger en blød klud til rengøringen.

Both are correct, but the first version puts extra focus on Til rengøringen.

It literally means for the cleaning.

In natural English, depending on context, it could be understood as:

• For the cleaning
• When cleaning
• For this cleaning job

The preposition til often means for or for use in.

Rengøringen is the definite singular form of en rengøring.

• en rengøring = a cleaning
• rengøringen = the cleaning

Using rengøringen makes it sound like a specific cleaning situation or task.

If you said til rengøring, that would sound more general, more like for cleaning or for cleaning purposes.

Yes, absolutely.

That version is also grammatical and natural:

Hun bruger en blød klud til rengøringen, fordi der er meget støv på bordet.

The difference is mainly emphasis:

• Til rengøringen bruger hun ... = emphasis on the cleaning situation
• Hun bruger ... til rengøringen = more neutral word order

Because klud is a common-gender noun, so it takes en in the indefinite singular:

• en klud = a cloth

The adjective must match that form:

• common gender, singular, indefinite: blød
• neuter, singular, indefinite: blødt
• definite/plural: bløde

So:

• en blød klud
• et blødt håndklæde
• den bløde klud
• bløde klude

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

So Danish usually does not use en/et with it here:

• meget støv = a lot of dust

This is similar to English, where you normally say a lot of dust, not a dust.

Because meget is used with uncountable nouns, while mange is used with countable plural nouns.

So:

• meget støv = a lot of dust
• mange bøger = many books

Since støv is uncountable, meget is the correct choice.

Der er is the normal Danish way to say there is / there are when introducing the existence of something.

So:

• der er meget støv på bordet = there is a lot of dust on the table

By contrast, det er usually identifies or describes something:

• Det er et bord = It is a table
• Det er støvet = It is dusty

So in this sentence, der er is correct because it is an existence statement.

Because på means on, and dust is on the surface of the table.

Also, bordet is the definite singular form of et bord:

• et bord = a table
• bordet = the table

So på bordet means on the table.

In standard written Danish, a comma is normally placed before a subordinate clause, and fordi introduces a subordinate clause.

So the sentence is divided like this:

Til rengøringen bruger hun en blød klud, fordi der er meget støv på bordet.

That comma helps show that the second part gives the reason.

Yes. Fordi introduces a subordinate clause, and subordinate clauses do not use normal main-clause verb-second word order.

For example:

• main clause: Hun kommer ikke
• subordinate clause: fordi hun ikke kommer

In your sentence, the clause is:

fordi der er meget støv på bordet

This already looks natural because der er is the normal existential pattern, but the general rule is still that fordi starts a subordinate clause.

Because hun is the subject form of the pronoun, and here she is the one doing the action.

• hun = she
• hende = her

So:

• Hun bruger en klud = She uses a cloth
• Jeg ser hende = I see her

In your sentence, she is the subject, so hun is correct.

The Danish ø does not have an exact English equivalent.

A useful approximation is:

• make a vowel somewhat like the one in British English bird
• but round your lips

So:

• støv has that rounded ø sound
• rengøringen contains the same letter in the middle

It is better to think of ø as its own vowel rather than trying to force it into a normal English sound.