Case I: When B takes off, A and B are both aged 0 in A's frame.
1) When B passes A, B is 30 (given in your setup).
2) But B's clocks run at half-speed according to A, so A says that B has been traveling 60 years.
3) Therefore A is 60.
4) But B says A's clocks run at half speed, so B says A was born 120 years before the shooting --- that is, 90 years before B was born.
5) So B's story is this: "90 years before I was born, A was born. He aged at halftime, so on my birthday, he was 45. At that time I started my journey to earth, which took 30 years. During that time, A aged another 15 years, so he was 60 when we met. Then he shot me. I died at 30."
Your mistake: You said that "from B's frame, the bullet has to be fired by A when B is 120". That's not correct. The correct statement is "from B's frame, the bullet has to be fired 120 years after A was born". Since B is 30 at the time of the shooting, A must have been born 90 years before B.
Your bigger mistake: You assumed that two different problems (namely this one and the one you asked in your last post) have to have exactly the same answer. In the other problem, A and B were in the same place at the same time when both were born. In this problem they weren't.
Case II: When B takes off, A and B are both aged 0 in B's frame.
1) When B reaches earth, he is aged 30 (given in the problem).
2) According to B, A ages at half-speed. Therefore A is 15, not 60 as you supposed. The 15-year-old A shoots the 30 year old B. Game over for B.