Superman/Clark is a script puppet in the killing scene, which is General Zod's suicide by cop. Even as I watched it, it was obvious that it was just tacked on because Nolan/Snyder think that sort of thing is fun. It didn't bother me much because basically I wasn't taking it seriously (in a getting into the story sense.) I was taken out of the story the moment Zod reappeared as if by magic.
I didn't find Superman killing entertaining myself, so I readily concede that those who weren't distanced at that point could find the movie as a whole to fail. Rating a story without taking into account the ending is pretty much a professional TV critic prerogative.
The Kents knew the damn dog could be saved, which explains why they were unrealistically concerned to make the effort. The real plot shortcoming is, as somebody noted upthread, the notion that an underpass is a tornado shelter.
That said, Martha Kent had no guilt for urging Clark to save the dog, for not telling Clark to save her husband or for not affirming the decision to Clark afterward. In short, she basically did not function as a human being. This is typically Nolan of the past. But I believe an essential aspect of criticism is to acknowledge the positive as well. Lois Lane was a real woman character. Having at least one is a great improvement on Nolan's previous work!