Medicus dicit puerum morbo gravi non laborare.

Because after dicit (says), Latin often uses an indirect statement construction.

In English, we say:

• The doctor says that the boy is not suffering...

In Latin, instead of using that plus a normal finite verb, Latin commonly uses:

• accusative noun + infinitive

So:

• puerum = the boy as the subject of the infinitive
• laborare = to be suffering / to suffer

That is why puerum is accusative, not nominative.

For the same reason: this is an indirect statement.

After verbs of saying, thinking, knowing, perceiving, and similar verbs, Latin often uses:

• subject in the accusative
• verb in the infinitive

So instead of something like:

• Medicus dicit quod puer non laborat
  Latin much more naturally says:
• Medicus dicit puerum non laborare

So laborare is not the main verb of the sentence. The main verb is dicit.
The infinitive laborare is part of what the doctor says.

Yes. This is a classic accusative-and-infinitive indirect statement.

The structure is:

• Medicus dicit = The doctor says
• puerum ... non laborare = that the boy is not suffering...

More literally:

• The doctor says the boy not to be suffering...

That literal English sounds awkward, but it helps show how Latin is built.

So:

• puerum = accusative subject of the infinitive
• laborare = infinitive verb

This is one of the most important Latin sentence patterns to learn.

Because laborare can take the ablative to express what someone is suffering from, troubled by, or afflicted with.

So:

• morbo = from/with a disease
• gravi = serious, severe, agreeing with morbo

Together:

• morbo gravi = from a serious illness / with a serious disease

This is a standard Latin idiom:

• morbo laborare = to suffer from an illness

So the ablative here is not random; it is the case regularly used with this expression.

Non negates laborare.

So the idea is:

• puerum morbo gravi non laborare = that the boy is not suffering from a serious illness

It does not mean:

• the doctor is not speaking
• the illness is not serious

It specifically denies the action/state expressed by laborare.

Because non stands right before the infinitive, it is very clear that the negation belongs to that infinitive clause.

Here is a breakdown:

• Medicus — nominative singular
  doctor; the subject of dicit

• dicit — 3rd person singular, present active indicative of dicere
  says

• puerum — accusative singular of puer
  boy; the subject of the infinitive in indirect statement

• morbo — ablative singular of morbus
  disease/illness; used with laborare

• gravi — ablative singular of gravis
  serious/severe; agrees with morbo

• non — adverb
  not

• laborare — present active infinitive of laborare
  to suffer / to be afflicted / to be troubled

Because gravi is an adjective describing morbo, and Latin adjectives must agree with the nouns they modify in:

• case
• number
• gender

Here:

• morbo is masculine, singular, ablative
• gravi is also masculine, singular, ablative

So they match:

• morbo gravi = with/from a serious illness

Even though the endings are different from what you may expect from first/second-declension adjectives, gravis is a third-declension adjective, so its ablative singular form is gravi.

The sentence is perfectly normal Latin word order, but Latin is much freer with word order than English because the endings show the grammar.

Current order:

• Medicus dicit puerum morbo gravi non laborare

A more grouped way to understand it is:

• Medicus dicit / puerum morbo gravi non laborare

Latin could also rearrange parts for emphasis, for example:

• Medicus puerum non laborare morbo gravi dicit
• Puerum medicus morbo gravi non laborare dicit

The meaning would stay basically the same, though the emphasis might shift.

So word order helps style and emphasis, but the case endings do most of the grammatical work.

Not here.

You may know laborare from the idea of labor or work, and in some contexts it can mean to toil or to work hard. But with morbo or other expressions of trouble, it means:

• to suffer
• to be afflicted
• to be troubled

So:

• morbo laborare = to suffer from an illness

This is a very common shift in meaning with Latin verbs: the exact translation depends on the words used with them.

Sometimes later Latin, and occasionally even earlier Latin in some contexts, can use quod-clauses. But in standard classical Latin, after a verb like dicit, the normal construction for indirect statement is the accusative-and-infinitive.

So classical Latin strongly prefers:

• Medicus dicit puerum morbo gravi non laborare

rather than:

• Medicus dicit quod puer morbo gravi non laborat

A learner should definitely treat the accusative-and-infinitive version as the standard pattern.

The direct statement would be:

• Puer morbo gravi non laborat.
  The boy is not suffering from a serious illness.

Then, to make it indirect after Medicus dicit:

• puer becomes puerum
• laborat becomes laborare

So:

• Puer morbo gravi non laborat
  becomes
• Medicus dicit puerum morbo gravi non laborare

That transformation is a very useful way to understand Latin indirect statement.