Детям было весело.

Because this sentence uses an impersonal construction.

In Детям было весело, the children are not the grammatical subject in the nominative case. Instead, they are the experiencers of a state: it was fun for the children. Russian often marks that experiencer with the dative.

So:

• дети = the children (nominative)
• детям = to/for the children (dative plural)

This is very common in Russian:

• Мне холодно. = I am cold. / It is cold to me.
• Ему скучно. = He is bored.
• Нам было хорошо. = We felt good / We had a good time.

So детям here means something like for the children.

Because детям is not the grammatical subject.

This sentence has no nominative subject at all, so Russian uses the default past-tense form было:

• past tense
• singular
• neuter

That neuter singular form is very common in impersonal sentences.

Compare:

• Дети были весёлые. = The children were cheerful.
  Here дети is the subject, so the verb is plural: были.

But:

• Детям было весело. = The children were having fun / It was fun for the children.
  No nominative subject, so Russian uses было.

Here весело is not a normal adjective agreeing with a noun. It is a predicative word expressing a state or condition.

In this sentence, весело means something like:

• fun
• enjoyable
• merry
• cheerful

So Детям было весело means the children were having fun or it was fun for the children.

This use is similar to other Russian state words:

• холодно = cold
• жарко = hot
• скучно = boring / one is bored
• интересно = interesting

These often appear in impersonal constructions with the experiencer in the dative.

Not in the usual English-grammar sense of modifying a verb.

It looks like an adverb in form, and in other contexts it can function adverbially:

• Они весело смеялись. = They laughed cheerfully.

But in Детям было весело, it does not describe how someone did something. Instead, it names a state: it was fun.

So learners are often told that this is a predicative adverb or a word of state.

Because Russian does not need one here.

The sentence is built around a state:

• было весело = it was fun
• детям = for the children

So the meaning is complete without a nominative subject like они.

If you tried to force in они, you would usually need a different structure, for example:

• Они веселились. = They were having fun.
• Дети были весёлые. = The children were cheerful.

But Детям было весело is a different pattern, and it naturally does not use они.

These sentences are related, but not the same.

• Детям было весело.
  = The children were having fun.
  This focuses on their experience/state.

• Дети были весёлые.
  = The children were cheerful/merry.
  This describes the children as having a cheerful quality.

• Дети были веселы.
  Also means the children were cheerful, but this sounds more formal or literary than весёлые in many contexts.

So the key difference is:

• Детям было весело = how they felt / what their experience was
• Дети были весёлые / веселы = what they were like

In the present tense, Russian usually omits the verb to be in this kind of sentence.

So:

• Детям весело. = The children are having fun.
• Детям было весело. = The children were having fun.
• Детям будет весело. = The children will have fun.

So the present has zero copula, while the past and future need a form of быть.

Yes. Russian word order is flexible, though the most neutral version here is:

• Детям было весело.

Other orders are possible, but they change emphasis:

• Было весело детям.
  More marked; can sound literary or contrastive.

• Весело было детям.
  Also possible, with emphasis on весело.

For most learners, Детям было весело is the safest and most neutral order.

More or less, yes—that is a useful way to understand the grammar.

A very literal breakdown is:

• детям = to/for the children
• было = was
• весело = fun / merry

So literally: It was fun for the children.

But natural English usually says:

• The children were having fun.
• The children had fun.

So the Russian grammar is not the same as the most natural English wording.

No. By itself, Детям было весело does not describe a specific action.

It describes a state or experience:

• they were having fun
• they felt merry
• the situation was enjoyable for them

If you want an action, Russian would use a verb, for example:

• Дети веселились. = The children were having fun.
• Дети смеялись. = The children were laughing.
• Дети весело играли. = The children played cheerfully.

So Детям было весело is about how things felt, not about one specific action.