ll "ילדים שמחים" can barely work as a way to say "children are happy", but it sounds terribly artificial; the natural way would be to add a copula, and say ילדים הם שמחים. Note that adding some adverb changes it a bit: for "children are usually happy" I'd tend to say "ילדים הם בדרך כלל שמחים", but omitting the copula sounds reasonable "ילדים בדרך כלל שמחים".
So "happy children" is a lot more natural a translation to "ילדים שמחים". But there's a snag: Duolino presents it as a full sentence (period in the end, capitalized first letter in English), and in this sense it's not a complete sentence. I guess it can be a sentence as a short answer to a question, e.g. "What do you see in the picture? Happy children". But that's confusing. I'd make it look not like a complete sentence.
I typed "Happy children" because there is no "are". But some people here explained that "The happy children" should be "הילדים השמחים". Since there is no definite article (the/-ה), I can see no difference in the sentence "children are happy" and "Happy children". Is that right?
I think the problem with "Children are happy" is that it's a general statement, i.e. about all children, whereas both "The children are happy" and "Happy children" would normally be understood as a subset of all children or some specific children. So, the presence or absence of ה on ילדים tells you which one is meant.
It happened again! I write שמכים. But correct spelling is שמחים, right?! Why do I get no notification on my mistake?! How am I supposed to learn propper spelling like that? This greatly diminishes my trust. I don't want to have to open comments every single time to make sure I really did it right if Duo says I did it right :(