I thought zere was used mainly for animals and that ji was for people. Is there any difference in usage?
That is exactly correct. But žere can be used for people if they eat too much or eat it in a disgusting way. It is not a nice word when used about people.
What's the difference between kure and slepice?
Kuře is the young slepice. That (kuře) includes what you buy for meat in a supermarket. Slepice is what lays eggs.
A cockerel is kohout and the zoological name of the genus is kur.