"Do you prefer the big box or the small box?"
I would translate と as "or" in this context, since you are being asked to pick one.
Also, いい has quite a few possible translation. I'd be inclined to use "agreeable" in this case, although "good" does work.
So, "Big box or little box, which (of these) is agreeable?"
The sentence does not specify "you" or "do you" at all. You could translate いい as "prefer" or "prefered" but it is a stretch. There are better ways of expressing your preferences in Japanese but it would probably require higher level grammar than what Duolingo has covered to this point.
Semantically they are the same. Some people will tell you that there is a slight difference in magnitude (大きな meaning a little bit "bigger" than 大きい), but I have found no evidence of that in any grammar book. None of my Japanese friends and colleagues confirmed that either.
Grammatically, 大きな / 小さな are a bit special version of rentaishi (連体詞, so-called "na-adjectives"). Special in a sense that unlike other rentaishi you may only use them as adnominals but not in predicates (for historical reasons, apparently).
In other words, you may only say 大きな connected to a substantive, e.g. 大きな箱, but not at the end of a sentence, so something like 箱が大きだ is a no-go. Instead, say 箱が大きい（です）.
If you are using it to describe a substantive attached to it like in this sentence, apparently it always has to be な.. If you are using it with other markers to describe something then you apparently use い For example :
"a big box exists" 大きな箱があります "the box is big" 箱は大きいです
This is what I picked up, hopefully it's correct
The reason why が isn't acceptable here is that どっち is the subject of the sentence, not 箱, therefore どっち is linked to the predicate (いいですか). Multiple subjects are acceptable only in compound sentences, and this is not a compound sentence.
箱 can only be the topic, if you so desire, marked with は.
I have trouble figuring out which part of the English sentence is the subject, topic or object, and thus can't decide which of the three particles to use (が、は、を). I especially have a hard time with the sentences where は is not used at all. I always seem to want to put it after the first noun in the sentence.
If anything you'd keep the first instance of はこ (because the second can be implied then), but you can't just leave a な adjective hanging like that. You can substitute a noun for the particle の however, and say 大きなはこと小さなのどっちがいいですか http://www.guidetojapanese.org/learn/grammar/nounparticles#The_particle
と is used when making an exhaustive list (i.e. you are listing every option). も is an "also/too/as well", but not really in a list-type fashion. Maybe if the sentence was "There is a big box. And there is also a small box. Which do you prefer?" the も particle would be best.
The verb should be at the end in a Japanese sentence. In this case, the "verb" is the copula, desu. So desu ka needs to be at the end and the rest should appear at some point prior to that.
Generally speaking, Japanese is much more flexible about word order, compared with English, because the particles indicate grammatical relationships, rather than the word order itself. However, there are still some restrictions and common sentence patterns.
See if you still agree after reading this.
I know it's been a while (a decade or so) since I took formal Japanese language classes in school but this was something we did in like the first or second semester. I was taught to ask this more like:
大きな箱も小さな箱もどちら方がいいですか。OR 大きな箱も小さな箱もどちら好きですか。(Although to be perfectly honest I was also taught only the i-adjective forms of ookii and chiisai.)
It makes me happy that many exercises accept kanji and grammar constructs that are not taught in the lesson but this one does not accept most of the alternatives I had in mind - いいの？ - の方がいいの？ - の方が好きなの？ Are not accepted. Or perhaps they're unnatural or indeed wrong?
Here's what I understand from my learning. Sometimes words have different writing (different kanjis or here, the hiragana VS kanji) depending on the context or the grammatical rules. Here for this one, 方 is not used. It would be in hiragana. And, as you mentioned the objet at the beginning, の is not necessary at the end.