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.
幹 gān is the traditional version of 干. Traditional characters are used in Hong Kong, Taiwan, and most everywhere except in Mainland China. I personally use traditional characters because I learned Chinese in Taiwan. 干 is the simplified version. Duolingo is helping me learn the simplified characters now.
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).
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.