popping noise

Ford typically uses a five volt turn on for their factory amps.  Your Kenwood has 12 volts on this wire.  The simplest way to correct this is to put a 1.5k ohm resistor inline on the remote wire.  This may get rid of the pop.  If it does not, the sub signal it's receiving may be too strong for it.  You can try using an adjustable line out converter on either the front or rear speaker leads and see if that gets rid of the problem.

Other than that, you can do away with the factory sub, but if you still must have bass, install an aftermarket sub.  There are enclosures made by MTX and JL Audio that are vehicle specific, and will have a factory appearance.  If that is of no concern, you can buy a prefab box, build one yourself, or have one built.  If you go this route you will need an amplifier.  Some of the MTX Thunderform enclosures are amplified.

While keeping your current setup, you can also try installing a bass knob inline on the rca's that feed the factory amp.  That way you may be able to turn it down to a suitable level.  You may need a short length of rca cables to do this.