Translation:Please consult a dictionary.
Yes... and to translate it into English, "please" is one, if not the foremost way to express that. The sentence expected by Duolingo, "Look it up in the dictionary", does not sound particularly polite.
You make a good point that it's common in English (not unlike Japanese) to increase politeness by phrasing it as a question – "could you", "would you mind", et cetera; but the Japanese sentence we are given is not in the form of a question, so we would be taking some liberties to translate it as one... not to say that Duolingo has shied away from doing exactly that in the past.
If "a" or "the" isn't accepted, please do report it. But I don't think this is so much a problem with the course as it is a difference between English and Japanese. When I teach English in Japan, I tell the students, "open your textbooks", and then the Japanese teacher will say 「きょうかしょひらいて」. The teacher doesn't say あなたたちのきょうかしょ, because that's not typically how they speak. So the translation doesn't have to be "your", but it has the potential to be.
No, not usually. But what if you were one centimeter tall, standing next to a dictionary which towers over you, and your friend notices a spider on the cover?
OR what if in your group of friends there's one guy you all call "The Dictionary" because he knows a lot of obscure facts, and one night you realize none of you have his phone number?
But of course neither of those scenarios are what's happening here.
I'm not sure what you mean. Are you asking why this sentence is not translated as "open the dictionary", or do you mean why did Duo not choose a different sentence? For the former, the answer is simple: it's not what it says in Japanese. For the latter, I don't know. Why not "please buy a dictionary", or "please throw a dictionary"?