From the wikipedia article on question tags:
The English tag question is made up of an auxiliary verb and a pronoun. The auxiliary has to agree with the tense, aspect and modality of the verb in the preceding sentence. If the verb is in the present perfect, for example, the tag question uses has or have; if the verb is in a present progressive form, the tag is formed with am, are, is; if the verb is in a tense which does not normally use an auxiliary, like the present simple, the auxiliary is taken from the emphatic do form; and if the sentence has a modal auxiliary, this is echoed in the tag.
But then later on:
If the main verb is to have, either solution (does/has) is possible
Using this rule, group 2 and group 4 would both be correct. (As an AmE, I prefer group 2 with group 4 sounding awkward to a degree approaching incorrect, but I'm unsure about BritE)
Following the same rule, group 1 would be correct and group 3 incorrect as has/have is the auxiliary verb, and so it should be used in the question. However, as a native AmE speaker, this actually runs counter to my intuition as I would prefer group 3. I have a feeling this has to do with the 'have got' construction somehow affecting things.
Edit: updated because I should have read the whole thing
On usage, the Cambridge Grammar of English (p883) states:
The present tense form of have with got used for possession is more
than twice as frequent in spoken BrE as in AmE:
On formality, Swan in Practical English Usage (p230) states:
Got forms are especially common in an informal style. ... In very informal American speech, people may drop 've before got. I('ve)
got a problem.
Best Answer
At school in the UK in the 70's and 80's, we were always taught that "get" and "got" were very lazy and ugly-sounding verbs to use, and that there was nearly always a better alternative. So people of my generation would probably tend to prefer the "Don't I have" variations, at least in formal speech, even though they're all grammatically correct.