Suggested answer:
(1) Must expressing necessity/obligation:
So now I must find a new doctor.
Have to expressing necessity/obligation:
We have to find a key to the problem of work...
(2) In the past, must refers to certainty, e.g.:
He must have arrived on a day when the crop had gone down-river.
(3) Yes, must can also express certainty in the present:
It must be below zero, Albert said...
(4) In (1), where must expresses necessity, it is a case of root modality. (3), on the other hand, shows an example of must with epistemic modality, expressing the "speaker's judgement as to whether or not something is true."
Read about must and have to here
Terminology: epistemic modality, modality, root modality