My vote is that the birds are Flamingos - check out the beak. At the top of the stone - Flamingos with their heads under their wings. And see the little Flamingos above them? Yup, gotta be Flamingos...
And I've pretty much decided the mammal is a female lion (associated with a female deity later called Ishtar in that region.
Check this rhyton from KulTepe Note the feet and note the shape of the head.