Learn Farsi: دکتر سوختگی مرد را درمان کرد و پرستار زخم زن را. - The doctor treated the man's burn, and the nurse the woman's wound.

The verb درمان کرد (treated) has been omitted from the end of the second clause. This is called verb ellipsis. Since the action in the second half uses the exact same verb as the first half, Farsi allows you to drop it to avoid repetition, matching the English structure of 'the nurse [treated] the woman's wound.'

It feels strange to end a sentence with را. Is that normal when dropping verbs?

Yes, it is completely normal and required here. The word را marks the definite direct object, which in the second clause is زخم زن (the woman's wound). Because را must immediately follow the direct object it marks, it stays attached to it, which leaves it at the very end of the sentence when the verb is dropped.

Since the verb is a compound (درمان کرد), could I keep the non-verbal part (درمان) and just drop the کرد?

No, you cannot split the compound verb in this type of coordinate ellipsis. You must drop the entire compound verb درمان کرد. Leaving the sentence as ...و پرستار زخم زن را درمان would sound incomplete and grammatically incorrect to a native speaker.

How do we know it translates to 'the man's burn' and not 'the burnt man'?

The phrase uses the invisible Ezafe vowel to link a noun to its possessor. سوختگی مرد (sukhtegi-e mard) explicitly links the noun سوختگی (burn) to مرد (man), meaning 'the burn of the man'. If it were an adjective describing the man, the vocabulary and word order would be different, such as (the burnt man).