Will try without any mod, then with animal package and EAS (I usually like to play with them enabled too).
Rabbits: clover or hay pellets for first category, lettuce, apples, cabbage for the second one.
Geese, ducks, chickens: grains (wheat, barley, sorghum, millet, rye, spelt, triticale etc.) and root cuts: sugarbeet cut, potato cut, taro cut, carrot cut etc. (you can buy them from one of the buying station or shredd them like sugarbeet cut)