Actualy, the different between "писатель" (a writer) and автор (an author) is that писатель is a profession or a hobby. It is a man who writes something. And автор can only be of something. Author can be and a writer and a painter and a sculptor (e.g. an author of the book or an article or an image or a statue).
(My native language is Russian, so I'm sorry if I make mistakes in English :) )
That's probably because an "author" in English is in it's primary definition a "writer", while it seems that автор is something broader and quite different in many respects.
I'd go so far as to say that, insofar as автор and "author" could be considered cognates, they are, to a great extent, false-friend cognates. It seems that you'd have to qualify автор as автор книг = "author of books" (or some other writing) if you wanted it to mean the equivalent of the English "author".
That's a really weird phrase in English. You'd be more likely to encounter it as "the writing is childish", qualifying the writing rather than the author of the writing - and then probably in a psychological description of a patient or a teacher's description of a student's abilities.
Children is a noun, not adjective. You will need to add 's to the end to make it an adjective, since its function in this sentence is to describe "writer."
The way you have phrased it makes it sound more like the writer actually writes children. Consider, "a script writer," "a book writer." These people write scripts and books.
I feel like "children's writer"/"children's author" is not a term that we would actually use in English. However, there doesn't appear to be any phase with this meaning in English, which is tricky. The closest would be something like "writer of children's books" or "author for children". I'm guessing that "детский писатель" is a commonly-used term in Russia, is that right?
“Children’s writer” and “children’s author” are both common enough in English. I don’t know if it is regional or not, but I am familiar with both usages and for conciseness would personally use them over the terms you offered. Again, potentially regional?
I won’t be able to answer your question about usage in Russian though - sorry! :) Let’s see if a native speaker can let us know.
Same as in English, structures like "a student of a school", "a bag of pens" or "Ministry of Defence", use the modifier after a noun in Russian (poetry is the main offender here).
детский is an adjective; adjectives are placed before nouns they are attached to (again, poetry may use the inversion)
Your site needs to fix the idea that it considers the 'apostrophe s' is an effing typo. Either actually write out the word with the apostrophe included (for example, children's writer... rather than having 'children' and the 'apostrophe s' separated, or, fix the the damn prompt that I have a typo.
The word for a child is irregular in both English and Russian. The English word child does not take the regular suffix -s in the plural, it takes the otherwise archaic suffix -en. You are correct in saying that the apostrophe follows the suffix for the plural; however, in this case that means that it follows the -en.