Medica dicit puerum male dormire.

Medica is a nominative singular feminine noun, meaning female doctor or woman doctor.

In this sentence, medica is the subject of dicit, so it has to be in the nominative case. The ending -a tells you it is feminine singular here.

It is the feminine counterpart of medicus.

Dicit is the 3rd person singular present active indicative of dicere, meaning to say.

So dicit means he/she says.
Because the subject is medica, here it means the female doctor says.

This is one of the most important things in the sentence.

Puerum is accusative singular, not nominative, because Latin often uses an accusative + infinitive construction after verbs of saying, thinking, knowing, hearing, and similar verbs.

So instead of saying:

• The doctor says that the boy sleeps badly

Latin says, more literally:

• The doctor says the boy to sleep badly

In this pattern, the subject of the reported statement becomes accusative, so puer changes to puerum.

Because the clause after dicit is an indirect statement.

Latin usually expresses indirect statements with:

• an accusative subject
• plus an infinitive verb

So:

• puerum dormire = that the boy is sleeping / sleeps

If Latin used dormit, that would be a direct statement instead, not the normal indirect-statement construction after dicit.

No. In this sentence, Latin does not use a separate word for that.

English says:

• The doctor says that the boy sleeps badly

Latin expresses that same idea through the accusative + infinitive construction:

• puerum male dormire

So the meaning of that is built into the grammar, not represented by a separate word.

In a way, it is both, depending on how you look at the structure.

More precisely:

• the whole phrase puerum male dormire is the object of dicit
• within that phrase, puerum is the subject of dormire

So semantically, the boy is the one doing the sleeping.
Grammatically, the entire indirect statement is what the doctor says.

Male is an adverb, meaning badly or poorly.

It modifies the verb dormire, telling you how the boy sleeps.

So:

• male dormire = to sleep badly

By contrast:

• malus, mala, malum are adjective forms meaning bad

Since this word is describing the action of sleeping, not a noun, Latin uses the adverb male.

Word-for-word, the sentence is:

• Medica = the female doctor
• dicit = says
• puerum = the boy
• male = badly
• dormire = to sleep

So very literally:

• The female doctor says the boy badly to sleep

That sounds wrong in English, but it is normal Latin structure. A natural English translation is:

• The female doctor says that the boy sleeps badly
• or The female doctor says that the boy is sleeping badly

Latin word order is more flexible than English word order because the endings show the grammatical roles.

You know from the accusative + infinitive structure.

In puerum male dormire, the accusative noun puerum is the understood subject of the infinitive dormire. So the boy is the one sleeping badly, not the doctor.

If male were meant with dicit, the sense would be says badly, which does not fit nearly as well here.

Dormire is a present infinitive. In indirect statement, the present infinitive usually shows action happening at the same time as the main verb.

So Medica dicit puerum male dormire means:

• The female doctor says that the boy is sleeping badly
• or more generally, The female doctor says that the boy sleeps badly

The exact English wording depends on context, but the idea is that the sleeping is contemporaneous with the saying.

The direct statement would be:

• Puer male dormit.
• The boy sleeps badly or The boy is sleeping badly.

After a verb like dicit, Latin changes that into indirect statement:

• puer becomes puerum
• dormit becomes dormire

So:

• Puer male dormit
  becomes
• Medica dicit puerum male dormire