Translation:Why is this answer incorrect?
You know how in some animes, people go ちがう! to tell someone they're wrong? I don't think the literal meaning is always taken (like the phrase for I'm hungry) for most words. I think chigau can also be used to express differing opinions/answers. The question more accurately asks "Why is this answer different (from the correct one)?" I'm not a native, though.
Yes. It's part of the implied meanings that duolingo doesn't get into. Japanese is a very roundabout language that hardly is every direct. i. e. when you turn someone down the literal English translation would be something like, yes, but I have plans.
So, you'd say: Q: えいがをみませんか？ A: ええ、けどよやくがあります。
So, by saying it's different 違う(ちがう), you are more politely saying that the person is wrong. Whereas, if you frankly state the person is wrong, it can be perceived as being rude, depending on the familiarity between the people having the conversation.
The ん is actually a contraction of の, so what you are really looking at is 違うのですか, where の is the nominalizer, so the translation is something like "is it the thing that it's different?" (The thing is, it's different).
For a more in depth explanation of how の is used in this way, see Tae Kim. http://www.guidetojapanese.org/learn/grammar/nounparticles#The_particle_as_explanation
I think "How is this answer wrong" should be a correct answer/translation. To me, the question is "In what way (how?) is this answer wrong?" In English, "why" can serve this function, as well, with perhaps a slight change in meaning. Jisho lists どうして as meaning "how" before it lists "why". https://jisho.org/search/どうして. It says when どうして means "why", it usually means it in a "for what reason/purpose" type of "why".
I'll try to explain. The ん is a shortened の that can imply you are either prefacing information or expecting an explanation. Essentially, it changes statements of fact or questions by requesting or stating "shared" information with the listener (that is it says either we should both know this information, or I want to make this information known or I want an explanation). See japanese.stackexchange.com/questions/3349/how-is-the-のです-working-here/3685#3685 for various uses.
Here it is requesting more information. This may be a little confusing because without the ん the phrase still is expressing a request for information. See also http://maggiesensei.com/2010/09/08/request-lesson-when-and-how-to-use-んn-のです→んです）/ for an explanation on how the ん is increasing the emphasis of the question.