"The child is fine."
Translation:У ребёнка всё хорошо.
No, it's grammatically incorrect. Хорошо is an adverb and hence cannot modify nouns. Hence grammatically it should be either "Ребёнок хороший" (the child is good) or "У ребёнка всё хорошо" (everything is well with the child). In both cases the Russian equivalent of "is" is dropped from the sentence, but is implied grammatically anyway. So in the first case "хороший" is an adjective used to describe the noun (ребёнок), while in the second case "хорошо" is an adverb (well) used to describe the verb -- the missing "is".
The meanings of these two sentences are quite different in both Russian and English though.
Note that in general Russian is much stricter than English (esp. American English) in distinguishing adverbs and adjectives - the Russian analogues of "good" and "well" are always distinct words with distinct grammatical functions.
P.S. You can also say "Ребёнку хорошо", meaning "the child is feeling well". Literately this sentence should translate into something like "All is well to the child", with both the subject (all) and the verb (is) not explicitly present in the Russian sentence in this case.