разбираться в чем-то из рук вон плохо
This entire phrase is a hyperbolic expression used to emphasise the idea of "how badly someone does something", correct?
Its grammatical construction eludes me, though; I'm not sure how to parse the phrase -- how these separate words get to have the figurative meaning as a whole.
из рук = (getting) out of hand?
вон = out there?
плохо = badly
Also: What other verbs than "разбираться" are commonly coupled with this expression?
To me it breaks down like this: плохо as if everything falls из рук вон.
You may imagine an unprofessional or tired person, who can't hold his tools tightly and they keep falling out of his hands. So he effectively can't do his job.
Из рук вон means just out of hands. Вон is an old fashioned word and used to indicate direction "from here/inside to the outside".
This expression may be coupled with the name of virtually any activity
Он водит/шьет/работает из рук вон плохо
and also in the phrase like
Дела идут из рук вон плохо - The things are very bad.
Плохо is an adverb badly and «из рук вон» is an idiom meaning very It can be used with different verbs but only with this adverb.
Он учится из рук вон плохо. Дела идут из рук вон плохо. И готовит она из рук вон плохо.=совсем плохо, очень плохо.
Let's try it this way:
First, just like the other guys said: in the expression itself the adverb is "плохо" ("badly"), and the "из рук вон" is the amplifier. Another adverb used here is "слабо" ("weakly", in the sense of quality).
Also the expression might be used in the reduced form: "из рук вон", implying the mentioned adverbs.
Altogether the expression conveys the sense of deepest disapproval or lowest quality and is used with verbs denoting state, action, activity.
How the expression came to be? Without digging in the books and seeing as it uses the hand metaphor, it's very likely, just like @AlexVB suggested, that originally it meant the quality of something made by person who "can't hit a nail straight or keep a tool in a hand".
Compare "все из рук валится" -- literally "everything falling out of one's hands", "can't do anything right/can't make busy with anything, because of feeling very bad or low", and rude "(чьи-то) руки (растут) из ж*" -- "(someone's) hands (growing) out of one's arse"), the person is all left hands and thumbs, can't do/make anything right.
