1. Which of Jor-El's official duties did he hate?
Jor-El hated being Krypton's official executioner. Of course, by "execution" what they meant was putting the prisoner into suspended animation and shooting him into space. It was his distaste for this duty that led to him inventing the Phantom Zone projector (aka Punishment Ray).
2. When the young Jor-El was cramming for his exams, what trick did he use to maximize his study time?
Jor-El used the Time-Stretch Globe.
3. What invention of Jor-El's led to him being elected to the Science Council?
The Phantom Zone projector (aka Punishment Ray) resulted in Jor-El's election to the Science Council.
4. Jor-El was stuck in the Phantom Zone twice when the projector failed to bring him back. What two people repaired the projector on those separate occasions?
The first time, Lara repaired the machine, from which baby Kal had removed a necessary part:
The second time, Lex Luthor (temporarily reformed by a ray that Jor-El had invented) saved him:
5. When Jor-El first got married, where did he work?
Jor-El worked at a missile base.
David got part of #4 correct. Blaze contributed the other part of #4 plus #2 and #5. Michael Rebain gets #3 plus part of #4. I stumped my readers with #1, although Michael Rebain was close.