Det er vanlig at barna legger seg tidlig.

Here det does not refer to a specific thing. It is a dummy subject or formal subject, just like English it in sentences such as It is common that...

So:

• Det er vanlig ... = It is common ...

This is a very common Norwegian pattern:

• Det er viktig at ... = It is important that ...
• Det er fint å ... = It is nice to ...

Vanlig is an adjective meaning common, usual, or normal.

In this sentence, it comes after er, so it works as a predicate adjective:

• Det er vanlig = It is common / usual

A common confusion is with vanligvis, which means usually.

Compare:

• Det er vanlig at barna legger seg tidlig. = It is common/usual that the children go to bed early.
• Barna legger seg vanligvis tidlig. = The children usually go to bed early.

At means that and introduces a subordinate clause.

So the sentence is built like this:

• Det er vanlig = main clause
• at barna legger seg tidlig = subordinate clause explaining what is common

In English, that is sometimes omitted, but in Norwegian, at is usually kept in this kind of sentence.

Barna is the definite plural form of barn.

The forms are:

• et barn = a child
• barnet = the child
• barn = children
• barna = the children

So barna usually means the children, meaning children who are known from the context.

If you want to speak about children more generally, barn is often more natural:

• Det er vanlig at barn legger seg tidlig. = It is common for children to go to bed early.

Because å legge seg is a reflexive verb meaning to lie down or, very often, to go to bed.

So:

• legger = lays / puts
• legger seg = lies down / goes to bed

The word seg is the reflexive pronoun here, meaning something like himself / herself / itself / themselves, depending on context.

So:

• Barna legger seg = The children go to bed

Without seg, legger usually needs an object:

• Barna legger boka på bordet. = The children put the book on the table.

The infinitive is å legge seg.

Some useful forms:

• å legge seg = to go to bed
• legger seg = go/goes to bed
• la seg = went to bed
• har lagt seg = has/have gone to bed

So in your sentence:

• barna legger seg tidlig = the children go to bed early

Norwegian often uses the present tense for general truths, habits, and things that are normally the case.

So here:

• er = present tense of to be
• legger seg = present tense of to go to bed

But the sentence is not necessarily talking about what is happening right now. It is talking about what is generally usual.

That is why present tense is natural here.

Tidlig means early and works as an adverb here.

In a neutral sentence, adverbs like this often come after the verb phrase:

• barna legger seg tidlig

That is the most natural, standard word order here.

So the clause structure is:

• barna = subject
• legger seg = verb phrase
• tidlig = adverb

Yes. After at, Norwegian uses subordinate clause word order, not the normal verb-second pattern of a main clause.

In your sentence, you do not see a big difference immediately, but it becomes clearer if you add something like ikke:

• Barna legger seg ikke tidlig. = main clause
• Det er vanlig at barna ikke legger seg så tidlig. = subordinate clause after at

Notice that in the subordinate clause, ikke comes before the verb:

• at barna ikke legger seg ...

That is a very important pattern in Norwegian.

Yes, that is also possible.

Compare:

• Det er vanlig at barna legger seg tidlig.
• Det er vanlig for barna å legge seg tidlig.

Both are grammatical and natural. The difference is small:

• at + clause sounds a bit more like stating a fact about what happens
• for + noun + å + infinitive often feels a bit closer to English It is common for the children to go to bed early

So the original sentence is completely fine, but this alternative is useful to know too.