I am taking my dog on my thur this year, and have this has been the most difficult issue. He can lay with me in a double wide hammock. The only problem is that when I get in it, he slides to the lowest point, and isn't very comfy for both of us. Aother thing I have tried is to have him lay on my chest and hips, and have his head on my chest or on the side of the hammock. I am still working on this. I eventually broke down, and will be getting a Tarptent Contrail for him. I know it is big and expensive for just a dog, but it only weighs 24 oz and he will carry it. I may send it home during the warmer part of my hike, but I want something to keep him from the wind and rain.
So it really depends on the dog if he sleeps in a hammock with you. Every time you move, he will move. If you shift in the hammock, he will slide down and make it very hard to get situated. So my opinion would be a pad during the warmer months, and maybe a small tarptent or something for the cold months.