"Why are you not eating breakfast?"
That works as a more general question. But in Japanese, why-questions are often completed with んです because they are asking for an explanation. It's a contraction of のです／のだ and is very commonly used when confirming or explaining something. You can sort of see it as if the person asking the question is more interested in why at this moment the listener is not eating breakfast, rather than questioning them why they generally don't eat breakfast.
ん is a contraction of の here. So more formally, this sentence would be どうしてあさごはんを食べないのですか？My idea of the の here is that it's used to move the sentence forward into question form. If you omit it, it would sound unnatural. But know that の sounds pretty formal and, you usually use ん instead.
If you remove です you sound like an overbearing lout. ～ないか？shouldn't be used unless aware of the implications. Namely, that you are speaking from a position of rude authority. You could say 「どうしてあさごはんを食べないの？」and be eminently friendlier.
However, until becoming better versed in these intricacies, it is best to stick to the ～ですか？ form.
We agreed that it is better to say (perhaps because it's more polite or that it's the formally taught language order here) "朝ご飯をどうして食べないんですか？" Then Duolingo marked it wrong. A site calls their version 外国語としての日本語 -- Japanese as a foreign language. Certainly the feel of it when they say it is different. More like, "Why are you not eating at breakfast?" Strange that eating breakfast, the action, might be imperative to them either way . . .
Yep. that;s also how you learned your own language. Observe, then imitate. But if you prefer grammatical analysis first, there are many courses that take that approach. Choose whatever you think will work best for you. (Someitmes called inductive vs. analytical learning, I think)
I normally don't use question marks -actually no punctuation at all- in my answers, whether in the Jp course or other courses, and have never been marked wrong for doing so. Jp don't use question marks in their formal writing, instead they only end their interrogative sentences with the normal Jp full stop (。). however it does commonly appear in casual writing, or mangas.