Translation:The girl drinks neither beer nor coffee.
Yes you can say that. As a matter of fact that is a more common way to say it. ( i'm dutch, and in daily conversation the word noch is not been used very often. It is correctly, but a bit uncommon nowadays) In daily life people would say "het meisje drinkt geen bier en geen koffie". That is more common.
"The girl neither drinks beer nor coffee" isn't correct English because you're pairing "drinks beer" with "coffee", rather than "beer" with "coffee". They need to be the same parts of speech or parts of a sentence. So you could have "The girl neither drinks beer nor pours coffee" or "the girl drinks neither beer nor coffee". Can you see the difference? It's a common mistake that many native speakers make.
I'm guessing the second is correct because it means essentially the same thing as the more literal translation and is correct English.
I disagree in that, once the general population adopts a manner of speaking, it becomes part of the language. There is an implied "drinks coffee". Perhaps don't write the sentence that way on a college paper, but there is no real ambiguity in what people would assume you mean. Also it may sound less correct outside the USA.
In very recent times the use of words like nor are falling out of favour making nor and or interchangeable but formal guides may still reference nor making it another example of an easy way to tell a native speaker from a non native one, at leadt within younger generations.
Because that is not correct English: http://dictionary.cambridge.org/grammar/british-grammar/questions-and-negative-sentences/neither-neither-nor-and-not-either
You can use not and neither, though only when there is a separate clause (which is not the case in this sentence).