Medicus matrem in cubiculum ducit, puerum sanat, et post paucos dies puer iterum in horto ludit.

Because matrem is accusative singular, showing it is the direct object of ducit (leads).
mater would be nominative singular and would normally be used for the subject.

With in:

• in + accusative = motion into a place (into the bedroom)
• in + ablative = location in a place (in the bedroom)

So cubiculum is accusative because the mother is being led into it.

Because here in expresses location, not motion:

• in horto = in the garden (where he is playing)
• in hortum would mean into the garden (movement toward/into it)

So horto is ablative singular to show where the action happens.

They have different roles:

• puerum (accusative) = the object of sanat (he heals the boy)
• puer (nominative) = the subject of ludit (the boy plays)

Latin uses case endings (not word order) to show who is doing what.

Yes. In Medicus ... ducit, puerum sanat, the subject Medicus carries over to the next verb sanat. Latin often avoids repeating the subject when it stays the same across coordinated verbs.

All three are present tense: ducit (leads), sanat (heals), ludit (plays).
Latin often uses the present tense in storytelling as a vivid “narrative present,” even if English might naturally use past tense depending on context.

post is a preposition that takes the accusative when it means after.
So:

• post paucos dies = after a few days

Here paucos is accusative plural masculine to agree with dies.

dies is usually masculine and belongs to the 5th declension.
Its nominative plural is dies, and its accusative plural is also often dies—so the form can look the same even though the case is different. In post paucos dies, it’s accusative plural because of post.

Because grammatical gender in Latin doesn’t follow “real-world” gender.
Since dies is usually masculine, the adjective must match it:

• paucos (masc. acc. pl.) + dies (acc. pl.)

iterum means again. It’s an adverb, so it doesn’t change form.
It’s placed before in horto ludit to highlight that the boy plays again, but Latin adverbs can often move around for emphasis, e.g. puer iterum ludit or iterum puer ludit.

Latin word order is flexible because case endings show grammatical roles. The order here is very natural:

• Medicus first to set the topic (the doctor)
• objects (matrem, puerum) placed before their verbs
• the final clause starts with puer to shift focus to the boy

Different orders are possible, but they can change emphasis.

Latin punctuation varies by editor and time period; it wasn’t standardized like modern English. The comma here helps the reader see three coordinated actions: 1) he leads the mother in
2) (he) heals the boy
3) and after a few days the boy plays again

So it’s more about readability than a strict rule.