Learn English – “Show,” “shown,” and “showed”

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

It is definitely not common, and just as you suspected it is a play on the previous two examples in the list of "Travelers" and "Givers".

It doesn't quite perfectly follow a trend: Traveler, Giver, Artist -- because the first two use the English language -er suffix to suggest the "actioner" of a verb (an action), and "Artist" is not "Arter".

But this is the syntactical joke being made: A traveler is one who travels, as an artist is one who "arts".

But if the question is whether or not this is common practice in English, then no, it is definitely not.

The act of artistic expression is understood to be an act of creation or construction, to do or make or create something.

The common expressions are: an artist creates...; an artist makes...; an artist expresses...and other such phrase statements.

Whereas it is proper and common English practice to say: a traveler travels; a giver gives.