Hun fejer gulvet med kosten, mens jeg henter en spand med varmt vand.

Because gulvet is the definite form of gulv (floor).

• et gulv = a floor
• gulvet = the floor

In Danish, the definite article is usually attached to the end of the noun instead of being a separate word like English the.

For the same reason: kosten is the definite form of kost.

• en kost = a broom
• kosten = the broom

Danish nouns usually take:

• -en for many common-gender nouns
• -et for many neuter nouns

So in this sentence:

• gulvet = the floor
• kosten = the broom

Not quite, although it is the same preposition.

In this sentence, med has two slightly different uses:

• fejer gulvet med kosten = sweeps the floor with the broom
  Here med means using or by means of.
• en spand med varmt vand = a bucket with warm water
  Here med means containing or having.

So English also uses with in both ways, and Danish does the same with med.

Because vand is a neuter noun: et vand in grammatical terms, even though in practice you usually just say vand.

When an adjective describes a neuter singular noun in the indefinite form, it usually gets -t:

• varm = warm
• varmt vand = warm water

Compare:

• en varm kop = a warm cup
• et varmt rum = a warm room
• varmt vand = warm water

So the -t agrees with the neuter noun.

Because spand is a countable noun, but vand is usually treated as an uncountable substance noun.

• en spand = a bucket
• vand = water

You normally do not say et vand when you mean water as a substance. So:

• en spand med varmt vand = a bucket with warm water

This is similar to English: you say a bucket, but just water, not usually a water.

Henter often means fetch, go get, or bring back depending on context.

So jeg henter en spand med varmt vand suggests:

• I go and get a bucket of warm water
• and presumably bring it back

It is often more specific than English get.

Examples:

• Jeg henter kaffe. = I’ll go get coffee.
• Kan du hente børnene? = Can you pick up the children?

So henter often implies movement to collect something.

Yes. In a normal main clause, Danish usually follows subject + verb + other elements.

So:

• Hun fejer gulvet med kosten
• jeg henter en spand med varmt vand

That is the basic pattern.

Danish is a V2 language, which means the finite verb is usually in second position in main clauses. In this sentence, the subject comes first, so the verb comes right after it.

Mens means while.

It introduces a subordinate clause:

• mens jeg henter en spand med varmt vand
• while I fetch a bucket of warm water

Yes, it affects word order. After mens, Danish uses subordinate-clause word order, so you get:

• mens jeg henter ...

not a main-clause inversion pattern.

If the mens clause came first, the next main clause would show inversion:

• Mens jeg henter en spand med varmt vand, fejer hun gulvet med kosten.

Notice:

• not hun fejer
• but fejer hun

That is classic Danish V2 behavior.

Because Danish spelling traditionally uses commas to mark subordinate clauses, and mens introduces one here.

So the comma separates:

• main clause: Hun fejer gulvet med kosten
• subordinate clause: mens jeg henter en spand med varmt vand

In modern Danish punctuation, comma use can vary slightly depending on the comma system being followed, but before mens in a sentence like this, a comma is very normal.

Not naturally in ordinary Danish.

The most neutral word order is:

• Hun fejer gulvet med kosten.

That puts the direct object gulvet before the prepositional phrase med kosten.

Danish does allow some flexibility for emphasis, but Hun fejer med kosten gulvet sounds unusual and marked. A learner should stick with:

• subject + verb + object + prepositional phrase

unless there is a special reason to change it.

Yes, kost can mean different things in Danish depending on context.

Common meanings include:

• broom — the meaning here
• diet / food in some contexts

For example:

• en kost = a broom
• sund kost = healthy diet

So context matters. In med kosten after fejer gulvet, it clearly means with the broom.

It would be:

Mens jeg henter en spand med varmt vand, fejer hun gulvet med kosten.

This is useful because it shows Danish main-clause inversion after a fronted element.

Why fejer hun instead of hun fejer?

Because when the opening position is taken by something else—in this case the mens clause—the finite verb must still stay in second position in the main clause.

So:

• Hun fejer gulvet ..., mens jeg henter ...
• Mens jeg henter ..., fejer hun gulvet ...

Both are correct, but the word order changes.

The dictionary form is at feje = to sweep.

Fejer is the present tense:

• at feje = to sweep
• fejer = sweep / is sweeping

The same pattern appears with at hente:

• at hente = to fetch
• henter = fetch / is fetching

Danish present tense is usually formed by adding -r to the infinitive, though spelling details can vary a little from verb to verb.