Yes, it is a partitive article, but it isn't mandatory like in French; it's just that it's extremely hard to define where one should be used. "Vuoi zucchero?" feels incomplete, but it could be said in a colloquial context, while "vuoi zucchero o miele?" is perfectly normal in any context.
You can find two lessons on this issue on bussu.com :)
first: Il primo appuntamento
second: Trucchi per rimorchiare
Literally dello=di+lo, so "Vuoi dello zucchero?" is literally "Do you want of the sugar?" or a better translation "Do you want some sugar?".
Your sentence "Do you want some of the sugar" would be something like "Vuoi dello dello zucchero?" and this is not what you were asked to translate.
I think Duolingo needs to do a better job distinguishing between 2nd person singular and plural... It would help for them to say "you all" rather than "you", because it can sometimes get confusing and the lines get blurred. I know it accepts answers that say "you all", but it needs to urge it more in the beginning lessons so student can tell the difference.
The ‘do’ can be omitted in the English translation and a question mark placed at the end of the sentence, can it not? There is no Italian word for ‘do’ in the given phrase. It is common in .english to say both “do you want some sugar?” and to say “You want some sugar?” So why is what I wrote incorrect?