Thanks, everybody, for the replies.
I think I have a pretty good understanding now.
Here is what I think I know:
- The mysterious black "button" is actually a plastic cap that covers a 3.5mm A/V input. I know because I pried mine out.
- Why was it capped? This socket might be needed for a Ford OEM radio, but the Axxera radio has that jack integrated into the front panel.
- Why four pins? Right audio, Left audio, ground, and probably Video. I suppose the fourth pin could have been for a microphone input.
- Why was the cable unplugged from the back of the socket? why not?
- What is the function of the other red and black connector? I don't know and I don't care. But, I'm pretty sure it has to do with another feature of the Ford OEM radio.