I have questions.
1) So Kryptonians can be killed on Earth by a piece of Kryptonian metal? If that's the case why didn't Zod do that to Superman in Man of Steel?
2) Once this was apparent, why didn't anyone try this approach to kill Zod himself?
3) How were the Russians suppressing Kara's powers? Shouldn't she have been in a Kryptonite cage or something? Possibly inspired by Injustice, DCEU Wiki claims she was in a "red solar prison cell", but was there any evidence of this in the film?
1) He didn't want to kill Superman, he wanted him to join them. That Zod still felt bad about killing Jor-El and didn't want to kill his old buddies kid too.
2) Barry in this movie was the only one in the position to do so, and he was more obsessed with saving Kara than killing Zod. Her death was an unchangeable event though, meaning even if he killed Zod something else would kill her. So possibly he DID kill Zod but Kara still died another way.
3) It took little Clark years to build enough Solar energy to gain his powers. Adult Kryptonians get theirs fast because of their matured bodies, but it takes longer for kids. So they kept her locked up since childhood and she never was exposed to enough solar power in the first place.