From Practical English Usage by Michael Swan -
We can use nowhere at the beginning of the sentence for the purpose of emphasis, and then auxiliary verb will precede the subject.
Example -
Nowhere had I less expected to see her than in my house.
In your example sentence, Nowhere in the United States but in New Jersey is a negative adverbial. There is the subject and are is the verb. So if it follows the same pattern it will be something like that -
Nowhere in the United States but in New Jersey are there so many people per square mile.
[I don't know if we use there are in the blank, it would be incorrect, but surely the stress factor would be missing. And it's recommended to use the inversion there. So it's better to stick to that.]
They all appear grammatically correct to me. Really the only difference is that some of the sentences are using more words than is necessary to convey the same meaning. For example, sentences 1 and 3 could go without "there are".
I suppose one could argue that sentences 3 and 4 make it more immediately apparent what place you are referring to (i.e. "this country") which might have a positive effect on clarification.
Another thing to consider is that if this were somebody speaking - at least from my experience - they'd be more prone to say sentences 2 and 4 because there is a tendency to be less formal.
Best Answer
First answer: violence is uncountable (and therefore singular), so the verb is is, not are.
Second answer: for many English speakers (but not all), there is (usually in the form there's) has become invariable, and they use it even for a plural complement. Eg There's a lot of people in the square. Not everybody would say this, but many would, and most people don't notice anything odd about it.
Even among people who say there's so many people, not all would be comfortable turning it round, and would say Why are there so many people?; but some would by happy with Why is there so many people?
In any case, none of this second answer affects your question (I just mentioned it for completeness): any English speaker would say Why is there so much violence?