Welcome! I hope you'll find the advice you're looking for. Admittedly, I am no expert on these particular games, but from I understand, this generation of Data East games are nearly identical to the hardware used in William's System 11 games (except for a few hardware differences here and there).
So, for the flippers, start easy. Check all the fuses. The flippers should be fused in the backbox on the PPB board. If the fuses check out, check for 50 volts at the flipper coil. Do this part with a game started. Also, while you're under there, check the end-of-stroke switches. They should be normally closed.
If that all checks out, there's a relay that engages the flippers. It's enabled from a transistor on the CPU board. Looking in the manual, it's a little vague, but I think It's Q80--don't quote me on that though.
Same goes for your pop bumper. Honestly, if it smelled like it was burning, it might be too late. Check for continuity though. Anything less than 2 ohms is a short. Otherwise, I'll bet that it's another fried transistor that's causing it to lock on. According to the manual, that'd be either Q8, Q9, or Q11. Those are the transistors for the pops. Again, I'm not sure which transistor corresponds to the pop bumper you're having issues with.
Testing procedures can be found on pinwiki. Not sure if you've been there yet, but it's a fantastic resource. There's also a ton of guys here that are far more useful than I am, but I at least hope I can steer you in the right direction. Good luck!