With headlights off: Front running lights work. Turn signals work (front and rear) Brake lights don't work when brake is depressed. Rear running lights don't work. Intermittent/inconsistent rear backup lights. ...

With headlights off:
Front running lights work.
Turn signals work (front and rear)
Brake lights don't work when brake is depressed.
Rear running lights don't work.
Intermittent/inconsistent rear backup lights.
Depressing hazard light switch causes the rear brake/running lights to stay on.

With headlights on:
Same as above BUT turn signals don't work!

- headlights and hi beams work fine
- I've replaced the brake switch (on the brake lever).
- I've put in new bulbs, and new light fixtures (all external lights).
- I've put in a new hazard switch.

None of this has changed the behavior at all.

The electrical is fine for the radio, dash, interior lights, heat/ac, and engine/computer.

Any pointers on how to figure out these issues? Are they a single cause like a bad electrical ground? Or, are there several issues with the electrical system? And, how could something like turning on the headlights make the turn signals stop working (if it was just a ground)?

In an older truck like this, you do not have to worry about electronics. It is most certainly a ground problem. Clean all the grounds you can find. I would also make up some wires with clips on them to put on some temporary grounds from the battery (-) to the different body parts to see if that helps.

Yep, yep ^^^^^^^^^ almost 100% certainty it's bad ground(s). Start cleaning and sanding and buying new straps.