Audio Systems: Amp bypass, mercury cougar, pioneer head unit

Question
I have a 1988 Mercury cougar that has a premium system in it. I have read a few questions and it seems pretty obvious to me that i need to bypass the amp or purchase the factory system integration adapter for the car which is sold on cardomain, but my plugs do not have the one rectangle and one square as is in the picture. Both of mine are the rectangular shaped ones. Another thing is i ran all new speaker wire to every speaker and disconnected them from the factory amplifier thinking this may solve the problem, but it didnt. Its not the head unit which powers on and all because i took it to my buddys truck who also has a pioneer head unit and we plugged mine into his harness and it works fine. Just need some sort of advice or direction to go in, ive installed a few units in my day, but this one has me dumbfounded.

Answer
this sentence doesn't make any sense to me:

Its not the head unit which powers on and all because i took it to my buddys truck who also has a pioneer head unit and we plugged mine into his harness and it works fine.

Do you mean "and NOW the head unit DOESN'T power on in your car or what"????

also you say your buddy ALSO has a pioneer head unit, does that mean you have one too?  this is very confusing
first of all I'm guessing at what the real problems are here.  I think it's obvious to say something about your system doesn't work but you haven't said that. you said you need to bypass your amp but you haven't said why, you said it's pretty obvious to you, but its not obvious to me since you didn't tell me what the problem is. so I'm guessing maybe that's what doesn't work? but basically what I would do is if something's not working and you've already done the work of runing new wires from all your door speakers to your cd player location, just get yourself a new aftermarket cd player and attach the speaker wires that you ran and abandon all the old wiring.  THIS HAS TO WORK.  if it doesn't, all your speakers are completely blown witch I highly doubt.

If I guessed wrong on any of this let me know, I'll try to answer better, but if you do write back please make sure it makes sense and I don't have to guess at what you mean.