The grammatical subject in question is that of "compound subjects."
Relative to the example, "How do X, Y, Z, etc. do that?" is correct form. This compound subject is unambiguously plural.
To the extent that the abbreviation for "et cetera" can be the cause of some confusion, it is useful to remember that the "et" part of "et cetera" means "and". If we translate "cetera" as "the same" or "the rest," we have an unpacked sentence that reads "How do X, Y, Z, and the rest do that?" When you think about the sentence in this form, perhaps the incorrect alternative would seem less appealing. "How does X, Y, X and the rest do that?" would probably strike the ear of a advanced English speaker as obviously incorrect.
Robertd is definitely correct in noting that there are such things as compound singular subjects like "peanut butter and jelly." But, your example is definitely not one of them.