I'm very new to Illustrator, so please forgive me if this is a daft question. Trying to colour the chicken's feet black where they overlap the white cow's back. I've tried making the chicken a compound shape but a message pops up saying it can't do that. Same happens when I try to make a clipping mask. I'm certainly missing a vital step here. Any help, much appreciated.
In AI there is more than one way to cook an egg... or paint a chicken. I tend not to use live paint groups, so my answer is based on what I use.
I think your layer structure is getting in your way. I would get rid of the extra groupings (live paint, group, etc) if you don't need them. I would keep one of the chicken shapes (you have two as far as I can see in your layers window, the wing and the cow shape. You can delete the layers you don't need directly in the layers window.
After it has been simplified I would select the cow's back and the chicken together and using the pathfinder, select Divide. This will turn the image in 3 pieces: chicken top, chicken feet and "cow back minus chicken feet". I have separated the 3 pieces in this sketch to show you how it would work. I did not include the wing, for simplicity.
With the shapes cut this way you will be able to select each one of the shapes independently and fill them with whatever colour you want.
One final note, you might find that when you continue working on your art it might be handy to have kept the original shapes intact (chicken with feet and back of the cow without the chicken feet cut out). If you want to do so, then just duplicate the chicken and cow back shapes before using the pathfinder. You can hide them if you don't need them but keep them in case you need to do some masking, overprinting, etc.