Spark/ignition issues?

That would be the brown wire connected to the + side of the ignition coil I assume. Yes, it's 12V there. It's also 12V on the blue wire side too, that's connected to the voltage regulator. Well, close to it. I get more like 11.5V both sides. If the resistor blew open one side would be dead wouldn't it?
Yes, it sure would, so it seems as if the ballast is OK.

It does seem odd that the ballast is getting hot when you're trying to start the engine, as the ignition switch should only provide power straight to the coil, bypassing the resistor. Hmmm.... If there is a short somewhere in the ignition switch or wiring, I suppose current could run back through the resistor. I'll need to think about this.