The answer to my own question will not deal with all issues mentioned, but because there are no other answers I'd like to share my experiences.
I wasn't personally camping in Georgia, but I met people that were camping and I saw people camping.
Georgia still has an underdeveloped tourist base, most of the sleeping places are in hotels (big cities) and guesthouses/private rooms. But I saw a camping place advertised in Mestia (Svanetia) and I saw people camping outside a bar in Vardzia (so their backyard was a kind of camping place, you could say.
I met a few Israeli tourists that were trekking from Kazbegi to Tushetia and they said they'd slept most of their nights in tents and there were no security issues. But they camped only in places far from villages, when they were near some village, they've stayed there.
I also met one guy travelling from Uzbekistan to Georgia by hitchhiking and sleeping in a tent, and he said he had no problems or security issues, neither from people nor from wild animals or dogs.
But one thing you should always avoid is getting near flocks of sheep, because they are protected by very large and aggressive dogs, whose task is to protect the animals from everything including wolves and bears.