Your pup may have mild irritable bowel disease(IBD), which is not helped by foods that may set her off(hence the gas), as everyone above said switch to a grain free food or something like hills i/d or nutro sensitive stomach which has a skin and coat guarantee. i think you will find switching her to the right food will also fix her skin issue. Not saying she has IBD but it is quite a common problem that people miss if it is mild. Sometimes you just need to keep switching foods till you find the right one for your dog. However first thing is first, take her to the vet 
