RE moving animals, I will not fire at them. Just my choice, but it sure seems like a few folks out there are good at it. And thats great!
My theory though, is for the way and places we hunt, it rarely takes moer than a few seconds to grab some kind of rest, and I'll do that every time. Even if it costs me the game.
Just a different way, but if you practice for getting a rest as another poster noted, then its second nature and quick.
Of course I rarely shoot an animal unless bow or my old war musket, thats close. Generally all my shots are around 175 and further out, its mostly due to where and how I choose to hunt. Pretty common to take at least a couple of shots at 300-350 on a particular trail. While I'm pretty sure I could do that offhand on a good day on our deer, Its much more reliable to get a rest. And since I won't shoot unless I"m 200% sure, its about the only way I do it these days.
Alaska fall hunting is a bit different at times and I"ve made a few quick moose shots around 125-150 off hand with zero other rests around. But they are not small and standing still...