If one ignores the contraction here, then this could be mistaken for "Can we not stay at home instead?" which - besides being wrong - has the opposite meaning to the correct translation!
Would "Kan vi stanna inte hemma i stället?" accurately translate to "Can we not stay at home instead?"
You can hear some, especially young people, use this in Swedish, probably because of English influence. It would be kan vi inte stanna hemma with a stressed inte in that case, no one ever says kan vi stanna inte hemma, that would be like 'can we stay not at home'.
But in Standard Swedish, the way to say what you're thinking of would be by using some other construction. Kan vi låta bli att … or Kan vi strunta i att … or Kan vi slippa …
"Can't we stay at home instead?" Can be used as a response to someone's suggestion to go to, say, school, or the dentist's office.
And "Can we not stay at home instead?" being "Can we do something that is the opposite of staying at home?" Such as going to an amusement park, or cinema.
The first begs to stay home instead of going somewhere. The second begs to go somewhere that isn't home. I admit, there is a stress on the not which may carry all the meaning here.
I think you'd normally take the "instead" out if you want to convey the second meaning. "Can we not stay at home?" with that strong stress on 'not', could be used to ask to do anything that isn't staying at home.
To ask "Can we not stay at home instead" with that same emphasis is getting into a really specific conversation. I think it would have to be something like:
A: I'm too tired to go for a run today, let's just stay at home. B: I don't mind not running, but can we not stay at home instead [of running]?
I'm not sure anyone has ever said such a thing!
A long explanation, but what I'm trying to say is that I think the 'instead' in this sentence virtually excludes the meaning of "I would rather do anything except stay home". And I think the same applies in Swedish, with the 'i stället'.
My best guess for translating "Can we not stay home?" into Swedish is simply "Kan vi inte stanna hemma?" but I don't know if Swedes use 'not' that way.
Yeah, I think it's a case of stress changing the meaning. I definitely see what you mean now, but I can also see it carrying the same meaning as the contracted version if said in the right way. Though most people would probably just say "can't" in that case, the other sentence sounds a little archaic :)
If you say jo, it means that yes, we can stay at home. Since it's a negative question, jo effectively replaces ja as an answer, so you shouldn't answer by saying ja - though you would be understood, of course. If you say nej, it means you're going out whether you want to or not. :)
Hope that helps!