"The man did the damage" is perfectly good English. Your instinct was correct, but you chose the wrong verb (I know many languages don't make the do/make distinction as in English).
Is there something like "Make the damage" in English? I do know do damage, as well as cause damage. But in this particular case, cause is a bit more fitting as the best translation, because it's reffering and "emphasizing" (if we could call it emphasis) to the actual source of the damage caused. In this case, the man. ^.^
I have never heard "made" used in connection with damage. "He caused the damage", "he caused damage", "he damaged it", "he made a mess of it" (using "mess" as a euphemism for "damage" in certain contexts), but no, never "he made the damage"
We use "make" a lot in English. He made a mistake, he made a mess.. although I agree cause is a better fitting word.
You make a mistake and you make a mess, yes, but you don't really "make damage". I can't explain that logically, it's just how idiomatic usage is.
Is "caused" correct? If a man jumps into the road, and a car had to swerve to avoid him and crashes into a lamp post, that man has caused the damage but he did not do the damage himself.
If the man however beats the car with a tree branch, that man does the damage.
Caused as in "triggered the events that lead to the damage" would be προκάλεσε. Έκανε is "did himself".
That's what worries me, as on the Android app the task is to choose the words bubbles, and it says "caused" for "έκανε".
English sounds awkward people are much more likely to say The man caused the damage