"Please give me a knife."
But I think at this basic learning level, we should learn the classifiers. It's kind of teaching to say "I am eating apple" instead of "I am eating an apple." In a very rare instance, you might even be able to say "I am eating apple" without sounding too ungrammatical if, for example, you are trying to imply that the fruit you are eating is called apple, and not orange. However, that's a very rare case. Same here! I think for us beginners it is very important to learn the nouns with their classifiers.
Not necessarily. If you use English logic, then you are right, but there are languages where the article is not required in this case, indeed, using the article would slightly change the emphasis in the sentence. Just try thinking of 'apple' as a kind of food, being uncountable (just as you can eat 'bread' or drink 'milk' without an article). I hope this is helpful.