Could you use a Slayer exciter in place of a NST to power a spark gap Tesla coil? I want to build a high power Tesla coil but unfortunately do not have the money or outlet to buy/plugin a pole pig to get really high voltage. I'm wondering if this is possible and what are the pros and cons of this setup.
The Slayer exciter is a form of Tesla coil fed by a built in transistor oscillator. It's ready as is.
If you want to run a traditional separate Tesla coil you can build a Ruhmkorff spark inductor to generate the initial high voltage pulses. Get a secondhand car ignition coil and build a mechanical or solid state switcher to get repeating pulses like it happens in the car motor.
I have seen a mechanical construction where the iron core of the ignition coil was the electric magnet of the switching buzzer just like in the original Ruhmkorff's version. Read this.
Striping the coil enough without breaking it to get near enough the iron core is not trivial. An external switcher is much safer because the ignition coil is made motor vibration- and waterproof.