Regina pueris ridentibus respondet se risum eorum amare.

Because regina is the subject of the main verb respondet. In Latin, the subject of a finite verb is normally in the nominative case.

So here:

• regina = the queen
• respondet = answers / replies

That gives the basic frame: The queen replies ...

Because respondeo normally takes the person answered in the dative case, not the accusative.

So:

• respondere alicui = to answer someone
• pueris = to the boys
• ridentibus = laughing, agreeing with pueris

Together, pueris ridentibus means to the laughing boys or to the boys who are laughing.

An English speaker may expect something more like answers the boys, but Latin uses the dative here.

Ridentibus is a present active participle from rideo, ridere (to laugh). It agrees with pueris in:

• case: dative
• number: plural
• gender: masculine

So it describes the boys:

• pueris ridentibus = to the laughing boys
• or more literally, to the boys laughing

This is a very common Latin way to add descriptive action without using a full relative clause like who are laughing.

Because after verbs of saying, thinking, knowing, perceiving, and often reporting or replying, Latin often uses an indirect statement construction.

That construction consists of:

• an accusative subject
• an infinitive verb

Here:

• se = accusative subject of the infinitive
• amare = infinitive
• se ... amare = that she loves ...

So after respondet, Latin says literally something like:

• The queen replies herself to love their laughter

But in natural English, this becomes:

• The queen replies that she loves their laughter

Se is a reflexive pronoun, used when the subject of the subordinate statement is the same as the subject of the main verb.

Main subject:

• regina

In the indirect statement:

• se ... amare = that she loves ...

Since she refers back to the queen, Latin uses se, not eam.

If Latin used eam, that would normally suggest her = some other female person, not the queen herself.

In an indirect statement, the subject of the infinitive is put in the accusative case.

So even though se means she in English here, Latin treats it as the accusative subject of amare.

Structure:

• se = subject of amare, but in the accusative
• risum eorum = object of amare

So:

• se risum eorum amare = that she loves their laughter

Risum is the accusative singular of risus, a noun meaning laughter (or sometimes laughing, a laugh depending on context).

It is in the accusative because it is the direct object of amare:

• amare risum = to love laughter / to love the laughter

So in the indirect statement:

• se risum eorum amare = that she loves their laughter

Eorum is the genitive plural of is, ea, id, and here it means of them or their.

It modifies risum:

• risus eorum = their laughter / the laughter of them

In the sentence, risum is accusative because it is the object of amare, but eorum stays genitive because it shows possession.

So:

• risum eorum = their laughter

Most likely, yes. The natural reading is that eorum refers back to the boys:

• pueris ridentibus = to the laughing boys
• risum eorum = their laughter

So the sense is that the queen replies to the laughing boys that she loves their laughter.

Latin often allows this kind of pronoun reference without repeating the noun.

Latin word order is much more flexible than English word order because Latin uses endings to show grammatical relationships.

In this sentence:

• regina is recognized as the subject by its form
• pueris ridentibus is recognized as dative
• se is recognized as accusative
• amare is an infinitive

So Latin does not need a fixed order like English does.

The sentence could be analyzed in chunks like this:

• Regina = subject
• pueris ridentibus respondet = replies to the laughing boys
• se risum eorum amare = that she loves their laughter

The order is natural Latin, even though it does not match normal English order.

Yes. A participial phrase in Latin is often best translated into English with a relative clause or another fuller expression.

So pueris ridentibus could be translated as:

• to the laughing boys
• to the boys who are laughing
• to the boys as they laugh

The most natural English choice depends on context, but grammatically all are ways of representing the same Latin phrase.

It can mean either answers or replies, depending on context.

In many contexts, respondeo is best understood as:

• answer
• reply
• sometimes respond

So this sentence could be rendered in English as:

• The queen answers the laughing boys that she loves their laughter
• The queen replies to the laughing boys that she loves their laughter

Because Latin uses the dative with respondeo, English replies to often matches the grammar especially neatly.