I imagine SHIELD is sort of like the NSA, the people in the know know about it, but if you interview the random moron on the sidewalk you might get a lot of confused answers.
I think it had to be Ward who shot the dog. That is actual a tie back to National Socialism. It was the SS (I believe) who were given a puppy to raise during training and as part of their graduation had to strangle it. So "shoot the dog" is probably a standard HYDRA test. Since it was Garrett testing Ward, if Garrett shot the dog, why would he then trust Ward if he failed the test?
As far as redemption for Ward, what I would like to see is since Ward seems more loyal to Garrett than HYDRA, Garrett bites it and Ward goes independent. He can then become sort of a recurring grey area villain/ally for several seasons and slowly get his redemption. Or a bullet between the eyes.