Thanks! So, the idea was that the achievement pop-ups would include the exact goals (but they tell you what not to do, of course).
Here’s a hint for the third level in case you want to try again:
Third level specific hint
In the case of the third level, the achievements you can get are “Collect 5 coins”, “Finish the level without taking any damage”, and “Finish the level without killing any enemies”, so the requirement to win is to both take damage and kill the noomba, and of course collect fewer than 5 coins.
You can satisfy the first two requirements by letting the noomba hit you before you kill it.