The fact that you still have the bracket sitting in there means that there WAS an amp in there, so you do, in fact, have the premium sound package that I keep yammering on about. I guarantee you, if you dig around under the dash and keep looking, you WILL find an extra connector hanging around there looking all lonely-like and begging for a mate. Buy the correct wiring harness adapter so you'll have a reference for what kind of a connector you'll be needing, and keep hunting.
Only other possibility I can think of is if someone monkey-rigged the whole friggin' car and just clipped the whole damned connector off at the end for some reason, or Frankenstein'ed the car by piecing multiple interiors together (no offense to Frankenstang, of course

), but otherwise, there SHOULD be a connector back there that you need to wire your stuff to, and that will do the trick.
As for why the stock Ford radio works and the aftermarket one doesn't ... I dunno. But again, I had the exact same problem in my '89 notch, and buying/wiring in the premium sound adapter harness was all it took to get sound through the speakers. I think that, even though you have wires going to the speakers from the other harness, you still have an open circuit somehow with the other connector that the stock amp used being disconnected, and so it shorts the setup out (hence the "pop" you get) when you turn it on.
EDIT: Why are you running a separate wire for the power and ground? Dude, the stock wiring harness has your power, ground, and constant battery right on there. You ARE using a wiring harness adapter ... aren't you?