No they do recognize their independence at least tacitly with the full expectation that the new independent states will be pro-Russian. Having independent states as Russian allies is not an oxymoron despite what some might think. That doesn’t preclude those new states from having ties with the west it only assures that Russia achieves its goal of having pro-Russian neighbor states.
No, it support the goal of the Confederation of Mountain Peoples of the Caucasus to create a Caucasian caliphate stretching from the Black Sea to the Caspian. Then next place the “Kosovo Precedent” will be put into effect is Chechnya. The Chechens support the Abkhaz and South Ossetians and are eager to finish the job of exterminating the Christian people of Georgia which they started on Russia’s behalf in the early nineties.