2 cents on not falling.
This is balance and stability, right?
So we're looking for a wide stance and good weight transfer.
You can throw any shot without twirling or twisting your body—by keeping your hips in line with the direction of throw. Try putting all your weight in your back foot, even so far as to lift your front leg off the ground (like a pitcher), then drive down and forward. When your done it should be most of your weight on your front leg. Making that snappy and smooth...should generate significant power and little instability. Your body stays put. Low possibility of slipping. If foot traction is very low (ice vs snow), simply not lifting the forward foot off the surface (point your toe and lift up heel) should suffice.
At worst you've just propelled yourself forward in a controlled slide—bend your knees—a few inches.
I'd be more worried about walking between shots (if there really is ice) than throwing, short steps if that's the case.
