Translation:Have you finished your homework already?
In this sentence, おわる"finish" is an intransitive verb (i.e. cannot be used with an object). That's why the particle は is used rather than を. The transitive verb for finish is おえる.
Transitive and intransitive verbs are tricky in Japanese and usually only mastered by the advanced learner. For now, just keep this concept in mind. For more information, including a list of transitive and intransitive pairs, see: https://en.m.wikibooks.org/wiki/Japanese/Grammar/Transitivity
The problem is that transitive and intransitive verbs are related but independent verbs, there thend to be some patterns but there are many more exceptions. It's better to try to memorize and use them in sentences and remeber if it uses を(transitive) or は/が (intransitive)
The you is implied (you is the implied SUBJECT - the person doing the finishing). あなた is not necessary as the speaker is clearly asking someone if they have finished their homework - we can tell this because the verb has an active voice, not passive - therefore someone needs to be performing the action ie. the finishing (of the homework). If we translated the sentence as you have above then the verb would be passive, and しゅくだい would be the subject. Also, the verb is positive - not negative, so there is no "hasn't".
おわる is an intransitive verb, the subject is absolutely the homework: The homework is finished, the homework is performing the action of being finished. There cannot be an object with intransitive verbs. If you want the homework to be the object you would have to use the transitive verb おえる. While it wouldn't be wrong to translate this phrase as "Have you finished your homework" if we were translating a book or something and trying to make it sound natural (because that is the more natural way to say it in English), in Japanese they are a lot more comfortable talking about things from the perspective of inanimate objects.
もう here is a particle meaning "already." It's usually taught in conjunction with まだ, which if used in this sentence, would translate to "Are you still finishing your homework?" (the verb would change to おわっています）
もう translates to "already" or "not anymore." The verb has already finished.
まだ translates to "not yet" or "still". The verb hasn't finished yet.
You would use mada for your example, threadpiece. Mou is asking if the listener has already finished their homework, are they all done? Mada means not yet and usually accompanies a negative verb or an implied negative verb/outcome ie. densha wa mada? Is the train not here yet? Un, mada desu - yep, it's still not here. Whereas mou indicates something that's already done or implies that something might be already done - in this instance the speaker is asking if the listener has already finished their homework. Duolingo's accepted translation doesn't shed any light on what it is supposed to mean though.
Well, that's where I disagree with the translation. You can't put "mada" in that question; "mou" fits into it. "Mada" fits into statements with "yet". You can see two similar examples with mou/yet questions here. http://jisho.org/search/yet%20%E3%82%82%E3%81%86%20%23sentences
As far as I understand it, how one could translate these words depends on the sentence, specifically question or statement forms. What I'm trying to say is: - In English, if you ask if someone has completed something, you either make a simple question with present perfect, "Have you finished it?" or you can add "yet" to it: "Have you finished it yet?" -- "Yes, I have (already)." / "No, I haven't (yet)." (Using "already" in the question potentially adds other implications. "Have you finished it already? Wow, you're so fast.") - In this same type of question in Japanese, もう can be translated as "yet". (The concepts behind "yet" and "already" are indeed similar, but the sentence translates more naturally and simply with "yet".) - Hence, "yet" is not always connected to まだ. In a question, "yet" can also be connected to もう. In a negative statement, まだ is "not yet".
もう宿題は終わりましたか。Have you finished your homework? / Have you finished your homework yet? はい、終わりました。Yes, I have. / Yes, it's finished. まだです。No, not yet.
It is perfectly fine to say "have you already finished?" in English. I feel that you don't understand the Japanese and how it translates to English. When you use mou it's referring to something which has already been done, something already finished/completed. Mou has a positive, optimistic expectation that something is already done/completed. mada refers to something which is either incomplete or still to happen. So when you use mada in a sentence it implies a negative outcome - in English (if we were to use mada in the original sentence instead of mou) it would totally change the meaning of the sentence to mean still not finished yet? mada has a negative expectation that mou does not.
First you say 'yet' would be better in the original sentence - that would require mada and imply a negative out come - not yet, which is not the meaning that the sentence is trying to convey with mou at all. But then you say it should be mou because mada doesn't work because it means yet. You seem to be contradicting yourself - I'm not sure what you're trying to say?
It's asking are you already finished? The speaker is presuming that the listener has already finished their homework - ie. a positive outcome. If the speaker presumed that the listener hadn't in fact finished their homework then they would use mada and the meaning would change to - still not finished? implying a negative outcome.
"Finished already your homework" is ungrammatical, but you could say:
"Have you already finished your homework?"
"Have you finished your homework already?"
You could say "is your homework finished already?" and that would be grammatical, but it's also a different construction, in passive voice.