Børnene har hver sin dyne, men katten sover helst på min.

Because børnene is the definite plural form: the children.

The noun changes like this:

• et barn = a child
• barnet = the child
• børn = children
• børnene = the children

So this is also a good example of two important things in Danish:

• the plural of barn is irregular: børn
• the definite article is usually added as an ending: -ene here

It is a very common Danish pattern meaning one each or each their own.

So Børnene har hver sin dyne means that the children do not share one duvet; instead, each child has a separate one.

A useful way to learn it is as a chunk:

• hver sin / sit / sine

It is used when a group is involved, but the possession is distributed individually.

Because in the expression hver sin / sit / sine, the form matches the noun that follows:

• sin
  • common gender singular noun
• sit
  • neuter singular noun
• sine
  • plural noun

Here, dyne is a common gender singular noun, so Danish uses sin:

• hver sin dyne

Compare:

• hver sit hus = each their own house
• hver sine bøger = each their own books

So the key thing is that sin matches dyne.

Because the sentence is talking about one duvet per child.

With hver sin, Danish normally uses the singular when each person has one item:

• hver sin dyne = one duvet each

If you used a plural noun, it would suggest that each person had several of them.

Because the noun dyne is already understood from the first part of the sentence, so Danish can leave it out.

That works much like English:

• my duvet → mine
• Danish: min dyne → min

So in Danish, min can stand on its own when the noun is obvious from context.

Here min means my one / mine, with dyne understood.

Because the missing noun is dyne, and dyne is a common gender singular noun.

Danish possessives match the noun they refer to:

• min = common gender singular
• mit = neuter singular
• mine = plural

So:

• min dyne → på min
• if it were a neuter noun, you would use mit
• if it were plural, you would use mine

Helst means preferably, by preference, or most willingly.

So katten sover helst på min means the cat prefers to sleep on mine.

A useful comparison is:

• gerne = gladly / likes to
• helst = preferably / would rather / most of all

So helst gives a stronger sense of preference.

Because the cat is sleeping on top of the duvet.

Danish makes a clear distinction here:

• sove på en dyne = sleep on a duvet
• sove under en dyne = sleep under a duvet

So på tells you the cat is lying on the speaker’s duvet, not underneath it.

Because this is normal main clause word order in Danish.

In a main clause, the finite verb usually comes early, and adverbs like helst usually come after the verb:

• katten sover helst på min

Also, men is just a coordinating conjunction here, so it does not cause any special inversion.

That is why katten sover helst sounds natural, while katten helst sover would not be the normal neutral word order here.