The Russia-Georgia border at Verkhniy Lars is definitely open, although this could of course change.
I crossed from Russia to Georgia two weeks ago with an American passport and had absolutely no problem. I don't see why you would want to miss out on Georgia, it is a very beautiful country with some great riding and quite possibly the most hospitable people I met in my trip through Europe and Central Asia.
I was coming from Kazakhstan and passed Astrakhan, Elista, Stavropol, Vladikavkaz, Georgia. Like Tony said I would definitely avoid Dagestan and Chechnya, not just because of militant activity. If you're caught without the proper permits off the main roads (M29) you can be arrested and deported.
Beware the police - I've never seen so many police speed traps and checkpoints anywhere. To put it in perspective from Estonia to Baikal and then to the Altai I was stopped a grand total of two times, in the Caucasus I was stopped four times in one day! And twice they tried to extract money from me (they didn't get any), the only negative experiences with Russian police in over 10,000 miles of riding in the country.