"Please give me a knife."
Isn't the Chinese closer to "Please give me (the) knife."? I think the more accurate translation is: “请给我一把刀”？
It's possible to omit counters in Mandarin if the meaning is still clear
把 is a classifier for long/flat objects with handles (also for chairs). It would be appropriate if you wanted to specify one knife but you can drop that here.
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.
"請給我一把刀" should also be accepted. If you have to say "給我一個盤子" for "give me a plate," I don't understand why "請給我一把刀" wouldn't be okay.
When I type in PinYin, it puts the characters in for me. After typing 'qing gei wo dao' there are no characters for knife offered. If I type 'qing gei wo yi', then the character is offered. So why is 情给我一刀, not accepted.
1) You don't need to specify quantity in this case (see other comments in this thread)
2) If you do want to specify quantity you need the correct measure word, in this case 把 (ba). So your sentence would read 请给我一把刀.