I don't understand, why is куртку accusative case here, and since it is, why isn't которая?
«Кото́рая спра́ва» is a subordinate clause, which has its a grammatical structure of its own: its own subject (кото́рая) and its own predicate (спра́ва). It's basically a sentence in a sentence. Since «кото́рая» is a subject, we mark it with the nominative case, like we do with all the subjects.
You imply that you want the person to give, or sell that thing, or whatever. You just do not specify the verb. Spoken Russian might provide you a handful of such mindscrewing structures, like "Зачем нам в Россию?" or "Лучше бы вдвоём пошли".
Imagine an infinitive of some giving-like verb there: *«Можно мне куртку?»* → "Is it possible" to me a jacket?. Clearly, whatever that verb were, "a jacket" would be its object (at least, if it is some straightforward verb like "to give"), so you always use the Accusative when asking for things without any verb.
Also, you may place your order in Nominative, just listing the things you need without any additional words.
Not sure. I'd explain it as a sentence with «да́ть» dropped, but I'm not sure that explanation is correct.
I think it's the same construction as in the requests like «Стака́н воды́, пожа́луйста» you say to a waiter or «Два биле́та до Ми́нска» you say in the ticket office, i.e. accusative is used when asking for things.
Will you hand me the coat that is on the right?
Is there anything wrong with this variant?
I thought this meant something along the lines of "can I have a jacket that is right for me" (i.e.: "that fits me"). I had not gleaned the spatial context of the question (the one on the right as opposed to the one on the left). How would you translate my other interpretation ("right for me" or "right for my size")?
Perhaps "... куртку моего размера" or "куртку, которая мне как раз", the latter closer to your wording. Maybe "куртку, которая мне подходит" if you wish to be especially vague whether it is about size or about style and colours. Or "куртку, которая мне идёт" if you mean exactly that the jacket should look cool on you.
All of these except моего размера are, of course, purely theoretical. It is nigh impossible for a sales assistant to know what is right for you before you even try the thing on.