The rotten wood may be a little damp but you definitely don't want it any wetter than you can avoid. You don't want to steam the hide back into rawhide. I'm glad you asked, that depends on what color hide you want. Pine will give it a golden color, while oak will give it a dark brown color. Other hardnesses in between those two seem to fall somewhere within that midrange of colors.
Cedar works very well and leaves a nice smell.
Don't let the coals cause the wood to flame up. Cover them with your smoker and pack dirt around the bottom so they don't flame up, just smoke. Keep an eye on it, a flare up could cook and burn that hide with its heat and ruin it. You may even need to pack dirt around where the stovepipe and trashcan meet, if the seal is not airtight enough. You also want to make sure there aren't any folds in the hide that will keep smoke from getting into all areas of the hide.