I work at a taxidermy shop and for the older mounts we dry clean them. The new mounts we shampoo them. When they tan the hide they use an acid to pickle the hide and then they neutralize it, and then put it in the tan. Over time they acid rot. If you soak a hide or mount that is acid rotted it will turn into wet newspaper. the newer synthetic tans don't acid rot near as fast. We first blow the dust off them using compressed air, and then use white gas on a rag to clean them. go with the grain of the hair and wiggle side to side as you go. it might take a few times until you quit picking up dirt on the rag.
As for shedding, you can't really reverse that. The bugs have eaten the roots of the hair, however if there is a possibility of anymore bugs spray the mount with permethrin.