Voltage regulator smoking?

A couple of suggestions when you are fixing this. I believe yours is a 1966 Dart? That means the wiring is almost 60 years old. The manufacturer never thought it would still be on the road in 20 years, let alone 60. Yoru under dash pictures show a lot of repairs from over the years, But I don't see burned, melted or toasted wires. you need to find the two fattest wires that come from your firewall, one red and one black and trace them under the dash looking for heat damage.

If you are going to keep the car for a long time and depend on it to get you where you are going, I would invest in a new wiring harness, There are companies that make complete factory correct wiring and that would clean up all the various repairs under your dash. This isn't cheap - be ready of sticker shock, but it and the brakes should be your first order of business in fixing your car up.

I don't have an early A-body, but I think that you can update your voltage regulator in kind with a transistorized model. Chrysler stopped using the type you have in the late 1960s. Somebody with an early A please provide better info.

You did just the right thing in your post by supplying lots of pictures!