In appendicitis, would a burst appendix initially lead to milder pain only then to get bad again?

Yes: That's how it happens sometimes. There is an abscess in the appendix, and it grows bigger, swelling and causing pain. Then the abscess leaks out of the appendix and the pressure inside decreases (called a burst appendix), causing less pain. The infection now is in the general abdominal area and will grow, thus needing treatment to avoid death in some hours or days.