Which modern dialects of Russian still pronounce the genitive adjective ending ого as written, instead of ово? Donbass, perhaps? Is the sound closer to g or to h? Are students being compelled to use the standard Moscow pronunciation in school?
The dictionary prescribed standard is ово. Moscow pronunciation is not really a modern term today (you probably meant historical Moscow norm, but today the pronunciation is prescribed by dictionaries and everyone is expected to say ово both in Moscow and Vladivostok, students in all places are expected to be taught the standard, in fact parents would have taught them it before school already).
But there are instances when ого ending could be pronounced as written. This is in R.Orthodox church sermons and usages connected with church, especially when quoting Bible in Church Slavonic, for in Church Slanovic it is pronounced AS WRITTEN (basically everything is pronounced as written in Church Slavonic). Outside of church or historical (biblical) text context such usage would be regarded old-fashioned and sounding a bit funny. My guess is ого-ending is anachronism and reflects the Church Slavonic roots of Russian (earlier historical Russian usages).
As for Ukranian and South Russia dialects (the latter I hear often at work), the speakers would pronouce such word as другого as [druhova], where 1st Russian г becomes Ukraninan vocalized h, but there's actually no need for them to change Russian в in ого-ending, at least I hear that this sound is pronounced as all other Russian speakers do (that is oго as в).
I unfortunately can't answer the first and the last questions.
But i will remark that currently Donbass is not unique in this respect, they speak Russian with the same accent as the rest of Ukraine including -aвa-ing the -ого ending.
It's rather native Ukrainian speakers who speaking Russian would pronounce -ого ending as -оho because this is how it's pronounced in Ukrainian or villagers speaking heavy surjik as it's called, a Russian contaminated with Ukrainian pronunciation, where Russian words are pronounced as if they were Ukrainian, mainly through replacement of the closed Е and И with the open Э and Ы, of Г with voiced glottal fricative [ɦ] and interspersing speech with some most basic Ukrainian vocabulary.
I wouldn't call it a dialect, but rather a distorted form of language.
The memorable General Secretary of the Central Committee of the CPSU "the dear" L.I. Brezhnev was the one to pronounce -ого like -aho or even -aha, but he wasn't a Donbass native, he was from Dnepropetrovsk region