In essence they are the input voltages multiplied by their relevant gains being added, if we call the top input resistor R1, the bottom R2 and the feedback resistor Rf, substituting them for say 1K and 500R resistors and x for the unknown input voltage we end up with: (1v * (Rf/R1)) + (x * (Rf/R2)) ...

Remember that in that configuration the output will be trying to make the -ve input of the chip 0v. If the resistors were all the same size you'd have the nice simple equation where the two input voltages will equate to the inverse of the output voltage. I think as you have half the resistance on on...

I don't know about how much volt drop you'd get over that sort of distance on something as small as telephone wire, it may be you need some sort of electronic amplifier in there to overcome this.

Feedback is where you have a complete loop and it feeds back into itself i.e. a microphone near a speaker which causes a squeal. I don't see any reason why a 9v pp3 will cause this, nor why 6xAA doesn't. Both will provide DC as smooth as a baby's bum (the voltage is from a chemical source, not deriv...

What is the spec of the adaptor?
You say -9v in the title so it is making me think you may have your polarity incorrect. To get -9v with centre pin 0v you need to connect the positive of the 9v battery to the centre pin and negative to the outer barrel.

If you want a more on/off switch then consider a darlington pair transistor. The BC547 has a gain of 110 to 800 which will depend on the voltage at it's base-emitter and a collector current of 100mA. Darlington pairs have a much higher gain.

Just in general, volume control in amplifiers for sound is usually log. He'll just need to double check as a linear slider fitted where a log one ought to be won't work very well.

Also pay attention to wether it is linear or logrithmic, (lin or log), volume controls are usually log.

I don't think AC will work, AC is a constantly changing sine wave - iirc the high voltage pulse needed is create by an instantaneous drop in voltage causing a back emf.

An ignition coil and a relay, what is providing the 450 pulses? You may need an ignition amplifier to fire the 12v to the coil.