Translation:What are we going to do this weekend?
I'd say that 幹gan tends to be used more commonly in colloquial situations or when there isn't a very clear idea of what are you doing, as in more abstract. While 做zuo has a more concrete meaning
做饭 = make food (cook) 做作业 = do homework
幹活 = to work
As a thumb rule, I'd say use zuo when you don't know which one goes, but there are expressions using both.
In Taiwan I find "gan" has an informal and impolite nuance. I would never use it to a superior or in business. Mainland China may be different. They are have a reputation of being rude, but that is not necessarily a language issue.
"Gan" can sometimes sound vulgar as well--at least in the Mandarin spoken in Taiwan (where it can in some contexts be used in a similar way to the "f-" bomb in English). "Zuo" does not tend to have that connotation.
In simplified Chinese, 干 is 乾、干and幹。But the context can lead you to the meaning.
I wrote "should" instead of "shall" and it marked the whole thing wrong.
I was given the words "up to" and "to do" so I created the sentence "What are we up to this weekend?". Isn't my sentence and the correct one synonymous in meaning though?
As a general rule, 会 only means "can" for a learned skill (cooking, speaking English, playing badminton, etc.). From my experience, the vast majority of the time, 会 is used to mean "will" rather than "can." In this sentence, it is more clear that 会 is meant to be "will" based off of the context where they specified a future time (this weekend).
"Gonna" is indeed informal. Instead of saying "gonna" you should say "going to". The answers should be: "What are we going to do this weekend?"
But note that other people are saying the Chinese use of 干 for "do" is also informal so maybe that is a good match after all?
做 is probably better to use in this situation because we probably don't want to come off as being rude
In English it is perfectly fine to write "What will we do next weekend?" It is another way of saying "What are we going to do this weekend?" There is basically no difference between the meaning of both expressions.
I see no reason why it is rejected when the Chinese says 会干 which has nothing to do with "going . It's more like 'can do' if anything.
They're not always the same. If today it's Saturday morning, and someone asks "what are we going to do this weekend?" I would expect that they're talking about today and tomorrow. But if they ask "what are you going to do next weekend?" I would interpret that as the weekend following the current one.
In the same way, I think 这个周末 would suggest the current weekend and 下个周末 the one following. I'm not certain if this idea works the same way in Mandarin though, so perhaps someone with a better understanding could clarify this better.