Umut,
Russia - Georgia border gate is open. Control on border more careful only. The name of this border gate is "Kazbegi" or "Verhnij Lars". The nearest railway is located in Vladikavkaz. You can go to http://pass.rzd.ru/ and take all information about russian trains (English interface present)

Wars between Georgia and Russia aren't present. But the North Caucasus which is territory of Russia - dangerous enough place. I advise to came to Russia directly or through Ukraine, if you can.

The hitchhiking is a good way of travel across Russia. You can pass 1000-1400 kilometers for light day in the summer. You should know only a few Russian and at once warn, that you travel a free of charge.