I tried building Donkey Breeders and Pig Farms on several maps now, but they never "produce" any animals, even if they are fully stocked on wheat and water. Do I need something else for them?

Just a stupid question: are those houses occupied with a worker?
And didn't you accidentally stop the buildings' production, did you? (they have a white flag with red cross on them)
Other than that they should work, my version works fine.

Yes, the houses are occupied and production is not stopped :(

And did you try to totally delete the game, then downoad and install again?
Or I can try to send you my saved games if they would work properly.

I'm experiencing the exact same problem. My suggestion is to temporarily disable the other buildings that require wheat and then you will see donkeys/pigs being produced. After that just return everything the way it was.
Hope this helps :-)

You may wish to check your transportation priorities. I tend to make almost everything equal, as then things simply go to the nearest place, which makes placement a bit easier. Of course this won't be a solution if you're playing a map with very limited resources and are forced to do otherwise.

Make sure they are top priority and they have grain and water.