I have seen from my next-door neighbor how raising chickens and cows together are a perfect match. The chickens learn to stay away from the cows so they don't get trampled, but they pick through the feed that the cows drop, eating what would otherwise get wasted. Chickens will even pick through the cow manure looking for fly larvae, supplementing their diet with extra protein. With enough chickens, your fly problem will be reduced significantly. However, they can't dig deep through the stuff, so chickens aren't going to solve a growing manure problem for you.
What about goats? You are going to have a few more challenges mixing chickens and goats, but you will have many of the same benefits that you do with cattle when they share the same yard. Here are a few issues that you will want to think about when considering combining your flock and herd:
Goats are high-energy, curious, and adventurous eaters. While it may be OK for the chickens to pick through the alfalfa that you feed the goats, you don't want the goats to get into the chicken feed. Goats will abandon their own feed in favor of the higher grain content from chicken feed. If you feed the chickens where the goats can't get access, this should solve the problem. In addition, I doubt that the goats would bother the hen's eggs, but it is probably safest if the goats don't have access to these either.
Both goats and chickens can spread coccidiosis to each other, a parasite that comes from picking through their manure. It can be devastating to both kids and chicks. This is the biggest reason why chicks are often fed medicated feed as a precaution while they are under the biggest danger (3 to 7 weeks). From the Goat Handbook, M. C. Smith; Cornell U., Ithaca, NY, "...although most goats carry coccidia and will have positive fecal exams, normally only the young kids become sick with coccidiosis. Deaths and stunted kids result. Raising kids separately from adults, keeping pens clean and dry, preventing fecal contamination of water or feed, and, in some herds, continuous preventative medication are necessary to prevent the disease. It is neither possible nor desirable to completely eradicate coccidia from the adult goats. A low level infection with the parasite serves to keep these goats immune to the disease."