Of course, you can, MaShuuSan. Both "nước ép táo" and "nước táo" are accepted.
Same. Thats how I spoke until this app and asking my cousin who lives in vn.
The rule explaned in the grammar says : when there is "one"/mon, you have translate with"a", when there is nothing, you have translate to "the". When "a man" should be correct, then the rules are not followed.