Zod's motivation in the entire movie (or at least in the Kryptonian portion) was to save Krypton by any means necessarily, when he realizes that he was unable to do so after seeing the planet's remains, he opted for a goal that brought back his nostalgic sense of Krypton. The problem is, how could he ever assimilate with humans? Especially with the hubris Kryptonians have that they're the more advanced race compared to humans. So in order to bring back Krypton, they resorted to taking control of Earth, and eliminating all unwanted populations, in essence Zod's motivations are similar to how colonialism worked in past history.
When Jor-El and Lara said that their fate is tied to Krypton, it's not because they wouldn't survive Earth's atmosphere (and frankly that doesn't make sense considering how much lighter Earth's atmosphere is compared to Krypton's, and the emergence of a young yellow sun). Instead, they say it because to uphold the sense of freedom that they want for Kal-El. Conversely, you could also interpret it as Jor-El and Lara sensing that their death alongside Krypton is their rightful punishment for breaking such a huge law and giving birth to a natural born baby.