Copiilor le este somn după cină.

Because this sentence uses the children as the experiencers of the state, not as the grammatical subject.

• copiii = the children in the nominative/accusative
• copiilor = to the children / for the children in the dative

Romanian often expresses feelings or bodily states with a dative person plus a fi:

• Îmi este frig = I am cold
• Îi este foame = He/She is hungry
• Copiilor le este somn = literally To the children, sleepiness is

So copiilor is correct because this pattern requires the dative.

Le is the unstressed dative clitic meaning to them.

It refers back to copiilor and repeats it in clitic form. This is very common in Romanian and is called clitic doubling.

So:

• copiilor = to the children
• le = to them

Together, Copiilor le este somn is literally something like:

To the children, to them is sleepiness.

That sounds redundant in English, but it is normal Romanian.

Because the verb does not agree with copiilor here.

In this construction, copiilor is in the dative and is not the grammatical subject. Romanian treats e somn / este somn as an impersonal expression, so the verb stays in the 3rd person singular:

• Îmi este somn = I am sleepy
• Ne este somn = We are sleepy
• Copiilor le este somn = The children are sleepy

So even with plural people, you still say este, not sunt.

Literally, somn means sleep.

But in the expression a-i fi somn, it means to be sleepy / to feel sleepy.

So:

• somn alone = sleep
• îmi este somn = I am sleepy
• le este somn = they are sleepy

This is a fixed and very common Romanian way to talk about sleepiness.

Yes, it is one of the most natural and common ways.

Examples:

• Mi-e somn. = I’m sleepy.
• Îți este somn? = Are you sleepy?
• Copiilor le este somn. = The children are sleepy.

You can also use an adjective:

• Copiii sunt somnoroși. = The children are sleepy / drowsy.

But le este somn is extremely common and often sounds more idiomatic for a temporary feeling.

Yes, that is grammatical, but it is slightly different in feel.

• Copiilor le este somn după cină = a very natural way to say they feel sleepy
• Copiii sunt somnoroși după cină = they are sleepy/drowsy using an adjective

The first version is often more idiomatic for a temporary physical state. The second is also correct, but it can sound a little more descriptive.

Because of how the plural and the definite article combine in Romanian.

Here is the breakdown:

• singular: copil = child
• plural indefinite: copii = children
• plural definite: copiii = the children
• dative/genitive plural definite: copiilor = to/of the children

So copiilor is not random spelling; it comes from the plural base plus the dative/genitive ending.

A learner often notices lots of i in words related to copil, and that is normal.

Because meal names are often used without the definite article when speaking generally.

• după cină = after dinner
• după prânz = after lunch
• după micul dejun = after breakfast

If you mean a specific dinner, you can use the article:

• după cina de ieri = after yesterday’s dinner
• după cina aceea = after that dinner

So după cină is the normal general expression.

Yes. Romanian word order is fairly flexible, although some parts must stay in the right place.

These are all possible, depending on emphasis:

• Copiilor le este somn după cină.
• După cină, copiilor le este somn.
• Le este somn copiilor după cină.

What usually stays fixed is that the clitic le comes before the verb:

• le este

So you cannot normally separate that pattern in an English-like way.

Yes, it is perfectly natural.

A Romanian speaker would easily say:

• Copiilor le este somn după cină.

A few close alternatives are also common:

• După cină, copiilor le este somn.
• Copiilor li se face somn după cină. = The children get sleepy after dinner.

So the sentence is standard, idiomatic Romanian.