There are two boxes of nails in the room. Left box and right box. There are also two men in the room. Me and my friend. How should I ask my friend to give me all the nails from the left box? My question is about articles. Also, I want to use attributive nouns. So: "Give me the left box nails." or "Give me left box nails."? I can make a guess that the first variant is more correct. But I want to know what rule applies to this case.
Of your two example sentences, the first one is better. Leaving out articles is certainly understandable, but sounds unnatural. "Left box nails" without the article would imply there are a certain class of nails called "left box" ones, and you want some. The article demonstrates that you want a specific set of nails, the ones in the left box. (Besides, although "left box nails" makes grammatical sense, I don't think there are any such thing.)
However, I wouldn't use either sentence. I'd use one of these:
Give me the nails in the left box.
Hand me the nails in the left box.
Please hand me the nails in the left box.
Could you hand me the nails in the left box?
The first sentence is absolutely correct, but "give" here sounds a little forceful. "Hand" sounds more like a request; adding "please" and turning into a question make it more and more polite and pleasant.
Finally, instead of "hand", if the situation is at the dinner table and the object is something on the table, "pass" would generally be used instead of either "give" or "hand." The successive modifications still apply.