1. My primary objection isn't even to the idea of Superman killing Zod, it's to the idea that Superman killing Zod is the only way to explain why Superman would never kill again. That's... basically saying the only way to know you shouldn't murder is to murder. It also taints the "Superman-as-Jesus" theme they keep using in Snyder's films, since suddenly you have a messiah figure who has himself engaged in "sin."
I'm not sure it was really meant to be that simplistic. For me the point of the scene wasn't to provide Superman with a lesson in how killing is wrong, but to show us just how painful it was for him and how far removed it was from his basic nature, and to show that this is NOT the kind of action he takes lightly (like so many other superheroes out there who don't react nearly as emotionally as he did after killing the bad guy at the end).
It seemed clear to me that he was being pushed to do something he absolutely did not want to do; it wasn't like killing was something he just routinely did beforehand, until he finally learns at the end that he shouldn't. I'm sure it did cement in his mind that killing was something he never wanted to do again, but I don't think he really needed to be taught or shown that it was wrong in the first place.