"아이들이 시끄럽게 놀아요."
Translation:The children play noisily.
I would completely agree if they used 아이는. But this sentence uses 아이들이, which is using children in plural form 들 and specifying them as a subject rather than a topic with 이. So the sentence is referring to a particular group of children, hence "the children" instead of "children" in general.
Not really. Native english speaker and "the children play noisily" sounds fine while "the children are noisy playing" I dont think is even grammatically correct. "The children are noisy while they play" is fine but a different sentence. I cant remember the korean but that would be "the children [while playing] are noisy"