Georgia will be united. Although the Georgian military was in a sorry state, I think that without Russian forces supporting the Abkhazians and South Ossetians they hardly wouldn't stand a chance. Things really depend on a 1991 or a 1993 POD (in 1993, the Georgian Civil War will already be in full swing, and Shevardnadze will have replaced Gamsakhurdia).
I would say that South Ossetia is toast, as they don't have nowhere near Georgian numerical strength and wouldn't inherit a lot of Soviet Army equipment. However, Abkhazia case is more complicated. I would say that IOTL Russian bumbling and half-assed support of Abkhazian separatists was much hampered by necessity to pretend to be non-aligned. None of warlordships of disintegrating Russia would encumber itself with such diplomatic niceties. What's more important, would Russia disintegrate it would leave sizable former Soviet Army units in Abkhazia (IOTL re-christened "Russian Army" and ordered not to get involved directly) without central lead and free to look after their own affairs. Abkhazian army was always multiethnic. Besides well-publicized Chechen involvement, there were units of local Armenians, Greeks etc., who were fighting to protect their homes, not to create Great Abkhazia. There're very high chances that orphaned Soviet Army would side with such a force in order to become one more group living on already multi-ethnic territory. Georgian side could not hope to gain such advantage. Georgian actions were fueled by very acidic and radical nationalism at this point and memories of 1989 "Tbilisi massacre" were fresh. They wouldn't "adopt" local Soviet Army (in fact, they would be very lucky if those units would remain neutral). Taking it all into account, I wouldn't be too surprised at "Abkhaz" army decisively beating Georgian paramilitary groups and "liberating" as much Georgian territory as they see fit.
Assuming that the separatists are defeated, I think Georgia will actually be more isolated. Most likely, Georgian para-military groups will have committed war crimes and where Russia might get away with such a thing, Georgia cannot.
C'mon, after Kosovo Albanians got away with what they did in 1999 and Croats got away with 1995 ethnic cleansings (or, if you want to look closer to region we're talking about, how about 1991-1994 Chechnya and expulsion of non-Chechen population), do you still harbour any doubts that a group's ability "to get away with it" depends on Western support for such a group? Western media coverage would be completely dependent on what group West would choose to support in this sorry mess.
Furthermore, in the aftermath of the whole conflict and assuming that many of the old opponents of Georgia will find refuge in the Northern Caucasus, Shevardnadze will have to install an authoritarian regime.
Saakashvili used violence to disperce peaceful opposition demonstration, sent masked gunment to smash sole opposition TV station into dust, blocked Georgian access to only part of Internet most Georgians are comfortable with and most likely rigged presidential elections of 2007. Did you see Western support of him waning because of it?
-Mostly Russian support of Armenia warded off a Turkish intervention in the Nagorno-Karabakh conflict. Without this, I think we'll see Armenia giving up on Nagorno-Karabakh. Without Russia and Armenia being so close, I think Georgia will take it's place, and the two nations will form a common front.
Yes, I don't see Armenia being able to stop Turkish attack, but that would be one nasty fight. However, I largely agree with you on likely endgame. Georgia (or what's left of it if Soviet Army helped Abkhazia) and Armenia (or what's left of it after Azeri-Turkish forces get Karabakh and whatever bits and pieces of Armenia they would see fit to "liberate") fighting shoulder-to-shoulder for their survival among militant Muslim nations. Essentially late 18th century with jets and machineguns.
-An independent Chechnya, and renewed struggles between the Ingush and the Ossetians. I think the Ingush-Ossetian war will be greatly expanded, with Chechen support for the Ingush, while greater amounts of Ossetians will join their brothers in the north.
-What do the Cossacks do? Create their own nation, or join a faction in the Russian civil war? What will their role be in the rest of the struggles in the Caucasus.
Cossack nations would not be created. Just forget about it. Cossacks were shooting a lot of wind about "separate Cossack ethnic story", but mainstream Cossack population never went as far as to declare Cossacks a separate nations. "Ethnic group of Russian nation" is as far as they would be ready to go (especially taking into account that urban population of said regions is "odds and sods" of every ethnic group from x-USSR, as usually happens with urban areas). Fate of Chechen and Ossetian-Ingush conflicts is hard to predict. Again, "shackles of civility" played a great role in Russian policy in region IOTL (even what is known as "ruthless Putin's campaign of 1999" was incredibly humane, as far as ethnic civil wars going, just ask Bosnians about it). Warlordships would likely not encumber themselves with silly notions of civility, so Northern Chechnya (flat and most populous area) would be pacified relatively quickly with blood-curdling measures. Picture wholesale destruction of villages within 5 miles of each and every guerilla attack, complete with machinegunning of population and livestock ("Belizarius"'s fans out there could remember Kungas exterminating rebellious Pathan clan; yes, this kind of things). Ossetians would likely be supported by said "Cossack warlordships", so Ingush would seat very nice and pretty after one or two very ugly events.
All in all, genocidal bloody mess, I'm not going to question it.