[JRTJH]

The first thing I'd check and verify is working properly is the battery. A dry battery can easily cause the converter/charger to overload, making it incapable of providing power to the lights, furnace, fans, etc. If there's a shorted cell or a bad cell or another issue with the battery, much of what you're experiencing can happen.

It may be something as simple as a corroded or loose battery terminal connection or a blown reverse polarity fuse. But all of those will "lead you right back to checking the battery"...