Der er brug for ro, hvis børnene skal sove.

Yes, in this sentence der works much like existential there in English:

• Der er brug for ro = There is a need for quiet

It does not mean a physical place here. It is a dummy subject used in statements about existence or presence.

Compare:

• Der er en bog på bordet = There is a book on the table
• Der er brug for ro = There is a need for quiet

So here der is grammatical rather than locational.

Der er brug for is a very common Danish expression meaning:

• there is a need for
• something is needed

So:

• Der er brug for ro = Quiet is needed / There is a need for quiet

A related expression is:

• at have brug for = to need

For example:

• Jeg har brug for hjælp = I need help

Even though brug often means use, in the phrase brug for it means need.

Because ro is an abstract noun here, meaning something like:

• peace
• calm
• quiet

In Danish, abstract or mass nouns are often used without an article when speaking generally.

So:

• Der er brug for ro = Quiet/calm is needed

If you said roen, that would mean the calm/the peace/the quiet, referring to something more specific:

• Ronen blev brudt = The calm was broken

But in your sentence, the meaning is general, so just ro is natural.

It can cover all of those ideas, depending on context.

In this sentence, ro suggests the kind of quiet calm environment children need in order to sleep. It is broader than just total silence.

So the idea is not only:

• no noise

but also:

• peace
• calmness
• settled surroundings

That is why quiet or peace and quiet is often a good English equivalent.

In this sentence, yes: hvis means if.

It introduces a subordinate clause:

• hvis børnene skal sove = if the children are to sleep / if the children are going to sleep

So the whole sentence has:

• a main clause: Der er brug for ro
• a subordinate clause: hvis børnene skal sove

In many contexts, hvis simply means if. Danish also has om, which can sometimes mean whether, so learners often compare the two. Here hvis is clearly the correct choice because this is a condition.

Because barn is an irregular noun.

Its forms are:

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

So:

• børnene = the children

This is one of the important irregular plurals to memorize.

Skal is the modal verb skal, which often means:

• must
• shall
• be supposed to
• be going to
• be to

In this sentence, it is not really a strong command aimed at the children. It expresses requirement/condition:

• hvis børnene skal sove = if the children are to sleep
• more naturally in English: if the children are going to sleep / if the children are to get to sleep

So the idea is:

• quiet is necessary in order for the children to sleep

Because hvis introduces a subordinate clause.

In the clause:

• hvis børnene skal sove

the order is:

• hvis = conjunction
• børnene = subject
• skal = finite verb
• sove = infinitive

That is normal Danish subordinate-clause structure.

A useful thing to remember is that Danish main clauses often follow V2 word order (the finite verb tends to come second), but subordinate clauses do not work in the same way.

Yes:

• Hvis børnene skal sove, er der brug for ro.

That means the same thing.

When the subordinate clause comes first, the main clause changes word order:

• er der instead of der er

This is because Danish main clauses follow verb-second order.

So compare:

• Der er brug for ro, hvis børnene skal sove.
• Hvis børnene skal sove, er der brug for ro.

Both are correct and natural.

Because hvis børnene skal sove is a subordinate clause.

In Danish, many writers put a comma before a subordinate clause like this:

• Der er brug for ro, hvis børnene skal sove.

That said, Danish comma rules allow some variation depending on comma style, and you may also see the sentence without that comma.

So for learners, the safest practical point is:

• the comma marks the beginning of the if-clause

Yes, absolutely.

• Der er brug for ro
• Der er behov for ro

Both mean that quiet is needed.

The difference is mostly stylistic:

• brug for is very common and everyday
• behov for can sound a little more formal or abstract

In ordinary speech, brug for is extremely common.

Literally, sove means sleep.

But in context, English often phrases this idea more naturally as:

• if the children are going to sleep
• if the children are to sleep
• if the children are to get to sleep

So although the Danish verb is simply sove, the practical meaning often includes the idea of being able to fall asleep and stay asleep, depending on context.