from where do i get the "we" in this sentence?
From the -elim ending, which is specific to "we"; ...elim! is "Let's ...!" and ...elim mi? is "Shall we ...?".
Is the way to say "shall I ..." some totally different thing?
Different yes, but totally different, no!
It's just -eyim /-ayım instead of -elim /-alım
Hastaneye gideyim mi? = Shall/should I go the to hospital?