Medica dicit puerum nunc domi dormire debere.

Because this sentence uses indirect statement after dicit.

In Latin, after verbs like dicit (says), the person or thing that would be the subject of the reported statement often goes into the accusative, and the verb of that reported statement goes into the infinitive.

So:

• direct statement: Puer nunc domi dormire debet.
  The boy ought to be sleeping at home now.
• indirect statement: Medica dicit puerum nunc domi dormire debere.
  The doctor says that the boy ought to be sleeping at home now.

So puerum is accusative because it is the subject of the reported idea, not the subject of dicit.

Because the reported statement itself already contains an infinitive construction.

The core reported idea is:

• puer ... dormire debet = the boy ought to sleep / ought to be sleeping

Here:

• debet = ought / should / must
• dormire depends on debet

When that whole statement is turned into indirect statement after dicit, the finite verb debet changes to the infinitive debere:

• puer dormire debet → puerum dormire debere

So:

• dormire = to sleep / to be sleeping
• debere = to ought / to have to in infinitive form, used because of indirect statement

It may help to think of the structure as:

• Medica dicit
  • [puerum ... dormire debere]
• The doctor says
  • [that the boy ought to be sleeping ...]

The subject of both is puerum.

In Latin indirect statement, the accusative noun often acts as the subject of the infinitives. So here:

• puerum is the one who dormire
• puerum is also the one who debere

In smoother English, that becomes:

• the boy is supposed to be sleeping
• the boy ought to be sleeping

So even though puerum is accusative in form, it still functions as the logical subject of the reported action.

Because Latin usually does not need a separate word like English that in this kind of sentence.

English says:

• The doctor says that the boy ought to be sleeping at home now.

Latin usually expresses that idea with accusative + infinitive instead:

• Medica dicit puerum nunc domi dormire debere.

So the combination of:

• puerum in the accusative
• debere in the infinitive

already tells you that this is a reported statement.

Domi is a special form meaning at home.

It is an example of the locative, an old case that survives in a few words, especially place words such as:

• domi = at home
• ruri = in the countryside
• some city names also use the locative

So:

• domi = at home
• in domo would mean more literally in the house

Those can overlap in meaning, but domi is the normal idiomatic way to say at home.

It can cover several related ideas, depending on context.

Debere basically means to owe, and from that it develops the sense to be obliged, to have to, ought to, or should.

In this sentence, the most natural English often is:

• should be sleeping
• ought to be sleeping

Depending on context, must be sleeping could also be possible in some sentences, but here should/ought to is probably the best match.

So puerum nunc domi dormire debere suggests that sleeping at home now is what the boy is supposed to be doing.

Yes. Medica is the feminine form, so it means female doctor or woman doctor.

Compare:

• medicus = male doctor or simply doctor in masculine form
• medica = female doctor

So the sentence specifically identifies the speaker as a woman.

Dicit is present tense: says.

The infinitives in indirect statement do not work exactly like English tense forms. Instead, their time is understood relative to the main verb.

Here:

• dicit = says
• dormire debere is a present infinitive construction

That usually indicates action happening at the same time as the saying. So the sense is:

• The doctor says that the boy ought to be sleeping now
• or The doctor says that the boy should now be sleeping at home

The word nunc also makes the time especially clear: this is about what is supposed to be happening now.

It most naturally belongs with the reported statement:

• the boy ought to be sleeping at home now

So nunc tells you when the boy ought to be sleeping, not when the doctor says it.

Of course, context can sometimes create ambiguity in Latin word order, but here the natural reading is that nunc modifies the boy’s sleeping/obligation, especially because it sits inside the indirect statement with puerum ... dormire debere.

Yes. Latin word order is flexible.

This sentence could be rearranged in various ways without changing the basic grammar, for example:

• Medica puerum nunc domi dormire debere dicit.
• Puerum medica nunc domi dormire debere dicit.

The endings show the grammatical roles:

• medica is nominative, so she is the one speaking
• puerum is accusative, so he is the subject of the indirect statement
• dicit is the main finite verb

Even though the order can vary, the given order is perfectly normal and clear.

The direct statement would be:

• Puer nunc domi dormire debet.

That means:

• The boy ought to be sleeping at home now.

Then, after Medica dicit, Latin changes it into indirect statement:

• puer → puerum
• debet → debere

So:

• Medica dicit puerum nunc domi dormire debere.

This is a very useful pattern to learn, because Latin uses it constantly after verbs of saying, thinking, knowing, hearing, and similar verbs.

Yes, and in this sentence that is probably the most natural English.

Latin dormire is just the present infinitive to sleep, but English often prefers the progressive form in context:

• to sleep
• to be sleeping

Because of nunc and because the sentence describes what the boy ought to be doing at this moment, English naturally comes out as:

• The doctor says that the boy should be sleeping at home now.

So the Latin form itself is simply dormire, but the best English translation may be to be sleeping.