The main issue here is the form to do someone wrong, and so on. Grammatically, this is a ditransitive construction: the verb takes two direct objects. Ditransitivity is more common with verbs such as give: “she gave me five dollars”, and so on.
Ditransitive do is certainly of quite limited use, but it’s hard to narrow down to just a few idioms. Searching Google Books for
did me a reveals many more examples of the construction than I’d thought of at first; but it does seem that it’s no longer forming new examples — in a quick flick-through, I didn’t find any new ones in the 20th century that hadn’t already appeared in the 19th, and some of the older ones seem a bit archaic now. But you can do someone at least:
- an injustice
- a favour
- a (dis)service
- a great honour
- a mischief (arch./dial.?)
- an injury
- a good turn
- a pleasure (arch.?)
- obeisance (arch.?)…
It brussed his body, & dyd him greate payne. —Bible (Coverdale), 2 Macc. ix. 7, 1535
To doo her ayde ayenst her enemyes. —Caxton, tr. R. Le Fèvre, Hist. Jason, 1477
There’s a second issue that’s slightly muddying the waters: the he done in for instance “He was my man, an’ he done me wrong.” This is an essentially independent issue: it’s just a non-standard past tense of do, incorrect in standard English but natural in some dialects.