My alternator wire melted.

Unfortunately new does not always mean good. Sorry to read about this.
Since turning the ignition off did not kill the power to the fire, at least one failure was the battery feed shorting to ground inside the alternator. Fusible link should have gone first, so maybe other things going on as well. (see example Fusible Links in Charging Systems with Ammeter)

On your alternator - regulator setup, the green wire goes to ground through the regulator. That's how those regulators control the voltage.
67dart273 I think covered the rest.