(Be) supposed to (pronounced /'spostə/, eye dialect spelling "sposta") is a periphrastic modal auxiliary predicate. Like ought to (pronounced /'ɔɾə/, eye dialect spelling "oughta"), it's a paraphrase for the modal should.
That means they have to be followed by an infinitive (which is why the to is there), and that one can substitute should freely for either be sposta or oughta,
- You're sposta empty the garbage.
- You oughta empty the garbage.
- You should empty the garbage.
at least in constructions where a modal is grammatical:
- He was sposta empty the garbage yesterday.
- *He shoulded empty the garbage yesterday.
The flexibility of such periphrastic modals in other constructions is why they exist. English speakers *would like to can use a modal as an infinitive, or in the past tense, the way German speakers can. But English modals are deponent, and don't have any inflected forms at all. So we use the periphrastic constructions instead in those cases.
There is not specific contrast between the used of sposta and do as auxiliaries; both take infinitives, and both can be used in questions of this kind, but so can many other constructions:
- What do I do next?
- What am I sposta do next?
- What should I do next?
- What must I do next?
- What do I hafta do next?
- What am I doing next?
- What does it say to do next?
- What's next on the list?
- What's the next step?
- What's next?
Use of sposta does not change the meaning of a question. Any of the questions above may be used either to simply request information, or to express irritation on the part of the speaker, but these would be distinguished by intonation and stress in speech, not lexical choice. In writing, which does not represent intonation, most sentences are ambiguous anyway.