EagerSeekerSo do you mean the "shall" is more polite way to ask than the "do" since it takes the other person's ideas into consideration with more interest? In that case, I guess I know what the similar expression is in my own language. :)
Not really. To me, it's not a more polite way of saying the same thing, it's a way of saying a slightly different thing that could be equally polite. (As it happens, prefixing your "do" sentences with "hey" does tend to make them sound rather abrupt, but this is a different issue from "do" vs "shall").
"Do" suggests an arranged, expected or habitual activity. In these circumstances, it's not rude to be not focusing so much on the other person's own ideas. For example, you're on a motivational training week. If you ask your companion "What shall we do this weekend?" then this implies you've got a free weekend and can choose to do what you like. If you ask "What do we do this weekend?" then this implies that something has been arranged. You don't know what it is, but you're hoping that your companion does.
Actually, since my original message, I thought of another occasion when "What do we do this weekend?" (and similar phrases) might well be used. This is when you're anticipating some sort of problem and puzzling over how you're going to resolve it. For example, you have lots of visitors in your house, and you've just about managed to find them all beds. Then you realise you're expecting your aunt to arrive on Saturday. You might say "Yeah, well, that's fine ... but what do we do this weekend?"