This is an interesting thing I've noticed about requests in Russian. I've been asked [/"ordered"] many times, "Девушка, закроете дверь, пожалуйста!" or even just "Девушка, закроете дверь/окно!", without the пожалуйста. In English, you preface requests like this with "Could you/can you/please". If someone just said to me, "close the door!" or even "close the door, please!", it would seem a little direct, if not quite rude! But this imperative form seems to be a perfectly acceptable and polite way of making requests in Russian.
So I do actually agree that "Could you take the cup please" should be an accepted translation; even if there is no explicit "could you" in the Russian sentence, it seems like the sense / level of politeness is roughly equal to that of a standard polite request in English, which would/should include "could/can you".
Well, in English many people consider "could you" as informal and "would you" as more correct and formal. "Would you close the window, please?" sounds perfectly polite and is a question. "Would you please close the window." would be the imperative command form. To that extent, adding the ending - те in Russian would be the equivalent as using "would you" without please. "Would you" is still more formal than "could you" and all commands ending in - те are considered formal. Of course, it's important to point out here that in Russian, the more words you use to say the same thing, the more polite it is. It is not unusual to excuse oneself excessively, i.e. "Извините, пожалуйста. Bы не подскажете мне, пожалуйста, станция метро - как прийти?" Albeit formal using the imperative with -те endings, you would always do well with adding a пожалуйста. "Принесите счёт, пожалуйста." You have a point that it is not necessary as you're already being formal, but I would consider it more polite. I am however neither English nor Russian, but Swedish. I would love for some native speakers to chime in and give their opinions.
Бы is only used with past tense verbs, and you could't really use пожалуйста in this case, so this would sound like “Взял бы ты чашку.” But somehow this doesn't sound very polite. :D It’s more like “I wish you took the cup.”
Another use of бы (and that is probably what you mean) is with “не мог бы ты” / “не могли бы вы” + infinitive: Не мог бы ты взять чашку, (пожалуйста)? This is basically like the English “could you…” in structure (to be more literal, “could you not…”), but it sounds way more polite in Russian than in English.
I'm not sure what you mean to say. It's true that the difference between "чашка" and "кружка" is a bit blurry these days, and I personally know people who use the word "чашка" for an object that's clearly a "миска" for me, but I wouldn't call them synonyms. Maybe that's regional usage.