"Where are we going tonight?" is a perfectly acceptable translation. I actually like it better than the one offered by Duolingo. I find, "Where do we go tonight?", although grammatically correct, a little stilted. If "Where are we going tonight?" was your answer, then I would report it.
This is frustrating. "stasera" means this evening so "Where do we go this evening" should be accepted. However "Where are we going this evening?" sounds much more natural, at least to an English speaker of English, than "Where do we go this evening" and should be accepted. Moderators please take note.
Yes. The Italian "questo" actually originates from the Latin "eccu istum" (check http://en.wiktionary.org/wiki/iste#Latin); separately, they originated the words "ecco" (here it is, there you have it) and "sto", which is a colloquial synonym for questo or its form in compounds. Thus originated "stasera" (this evening), "stamattina"/"stamane" (this morning), "stavolta" (this time) and so on.