The interrogative form present simple of all lexical verbs except BE, and, for a small number of speakers, HAVE, is formed with the auxiliary verb DO.
he works - does he work?
When HAVE is used as a lexical verb, its interrogative is formed with DO for most speakers; a small number of people simply invert subject and verb. The same is true of HAVE TO. When HAVE is used as an auxiliary, the interrogative is always formed with subject-auxiliary inversion:
He has - does he have? / (has he?)
He has to work - does he have to work? / (has he to work?)
He has worked - has he worked?
When HAVE GOT is used in place of HAVE, then the HAVE part of HAVE TO is an auxiliary verb, and the interrogative is formed with subject-auxiliary inversion. The same is true of HAVE GOT TO.
He has got - has he got?
He has got to work - has he got to work?