"Het meisje drinkt noch bier noch koffie."
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.
General consensus is that neither is paired with nor and either is paired with or. Hence in standalone sentences it is best to stick to that.
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).
"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.