I've recently become a hammock camper and I love it. Just be warned there is a real problem with a hammock in cold weather. You lose a LOT of heat from under you. You need a CCF pad between you and the hammock at least and maybe an underquilt. If you do that then it should be fine.

My guess is that your dog will just curl up under where you are hanging. That way he'll be protected from rain/snow by the tarp that you put over your hammock and he'll be close to you. No wind protection for the dog of course but that's what fur is all about.