You can only leave out the 'me' in this sentence if it shows up in a previous sentence or has come up in the conversation before, and even then, adding it is preferable to keeping it in, as without, the sentence sounds incomplete, informal and lazy. I'm with Theron on this one.
I will only use 'give the knife' (or item, whichever) when I'm talking to my most obnoxious of brothers, implying an informal setting, and this is a repeat request which he is willfully ignoring because he is a.. uh.. a brother.
It's not that "Give X" is never said in America. To me, at least, it's that it sounds rather rude, and like something which would never be combined with "please," unless the "please" was dripping with sarcasm or intentional irony. If пожалуйста wasn't there, I'd let "Give the knife" be added, but I just can't really imagine "Give the knife, please," being said with sincerity.
DL is confusing. Or does that mean that дай мне нож is wrong ? I can easily see mostly in movies... Give the knife or give that knife. Then I can consider more easily me with a knife than with the knife. Give a knife and give the knife isn't the same. But in Russian there is no article to make the difference..