Cut wood when you want to or need to, or when the opportunity presents itself. I prefer winter because it's easier to see, the weeds and underbrush are down, and it's not hot. And no bugs. But I have cut in August in Ohio and had no problems with the wood.
You need to remove bark and sapwood from osage at the same time in almost all cases.
Store cut wood in an unheated space until you get it down to near bow dimensions or a few months have passed. Green wood loses moisture really fast and this moisture loss causes shrinkage. If it happens too fast, the shrinkage near the surface or cuts ends separates the fibers and causes checks.