Here is a real Turkish song, full of this form for different verbs (it's also has lyrics written instead of any other action on the screen, "official lyrics edition"): https://vk.com/video72595_456239023 It starts with küsmeyelim, which is also an example of negation: 'let us NOT be offended/cross' (from küsmek 'to be offended/cross/vexed').
If "what will we do?" Isn't a correct translation.. how would you say that?
"Ne yapacağız?" (there's already a question word 'ne', so no place for the 'mi') asks for facts - "What are we going to do? What will we do?". It is the plain future.
Some conservative speakers of English use "shall" for the first person future ("I shall, we shall" but "you will, he will, she will, they will"), so they would say "What shall we do?" for this form.
"Ne yapalım?", on the other hand, asks for suggestions. This is the meaning I assign to "What shall we do?". Another translation might be "What do you think we should do?" or "What do you suggest we do?"
Ah, I see; thank you. The distinction in English between them seems rather hazy (to me, 'what will we do?' sounds more like it's asking for suggestions, 'what shall we do?' sounds like simple future, and 'what are we going to do?' could be either).
Oh, and thank you for reminding me that 'mi' is for yes/no questions; the recent slew of new suffixes has given me mental indigestion.
First question: see the thread started by RustyBucket16.
Second question: because of the -alım/-elim ending which is specific to "we"
Third question: there is no stand-alone word in Duo's Turkish sentence that corresponds to "we". It's clear from the verb ending so there's no need to say biz "we" explicitly as well.