I would say in the language I speak and hear daily there are three differences:
(1) In the affirmative, "must" is stronger than "have to".
(2) In the negative, "must not" denotes an obligation not to do something, while "does not have to" denotes an absence of obligation to do something, in other words leaves the choice free.
(3) The register of "must" is higher, more formal, than "have to", which should probably be avoided in formal writing. If you need something weaker than "must", say "ought to".
PS. All your sentences are correct. The last two do sound literary, very cultivated. In the first two the "must clean" is the stronger, as I have said. By the way, "have got to" is even more casual than "have to", and if you want to descend to illiteracy, you could, I suppose, omit the "have" and say "I got to". In this case the contraction "I gotta" actually seems less illiterate, for it clearly marks a careless pronunciation and little else.