I can share some personal experience from our trip around the black sea in september 2010.

Our route was Estonia-Latvia-Belorussia-Ukraine-Russia-Abkhazia-Russia-Turkey-Bulgaria-Romania-Hungary-Slovakia-Poland-Lithuania-Latvia-Estonia, some 8000km in 18 days.

No carnet needed, in every custom clearance bikes were treated as normal vehicules, no specific tax or duties.

From Russia to Abkhazia (Psou river crossing) with the prior confirmation received via e-mail from MFA Abkhazia is no problem with EU passport and plates. Unfortunately Abkhazia-Georgia (Ingur river crossing) is not possible with EU passport and plates (only at own risk in gargo of a local truck, we skipped this). The problem is Georgian authorities. We tried this both ways, through diplomatic channels all the way up in Tbilisi and "unofficially" on site. As Georgia is not recognizing Abkhazia, for Georgia, its border is still with Russia at Psou river. As normal foreigner entering Georgia needs to have passport control at the border. At Ingur river, you are tresspassing as it is 200km from the Georgia-Russia border. So you get arrested, can't blame them. I do not think you can do it from Georgia to Abkhazia, but have not tried this.

Safety issues, we had no problems and I do not think the risk in Abkhazia is higher than in Russia or Ukraine as an example. Local poverty and too much shining bikes and stuff can always create some issues. But nothing an adventure biker cannot solve.

When we had to turn back from Abkhazia to Russia, we had no choise but to take the ferry from Sotchi to Trabzon, to complete our tour around the black sea. However this is a robbery, GS + rider and a sleeping bag place (free) on the upper deck under the sky cost around 400USD. And the ferry cruises on "free schedule". An alternative is to find a gargo ship from Novorossiysk (Russia) to Batumi (Georgia), but there is even no "free schedule".