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.?)…
Edit: quotations in the OED show that this usage dates back at least to 10th century Old English (i.e. as far back as it traces the etymology), and was formerly used with some more words still:
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.