Your antenna problem is due to you having a Pioneer head Unit, as they only have one remote or trigger wire which is common for an amplifier and an antenna.
As someone mentioned, Kenwood head units have 2 wires, both are blue, one which is a constant power remote for the amplifier which is labelled P.CONT and the other a constant power which the head unit activates ONLY when the source is set to TUNER, (when your listening to the radio) this is labelled as ANT.CONT. The only way to solve your problem is a switch, or to buy a Kewnwood deck or just dont worry about it! But is it really that bad?
Another way to check your speckers, (assuming everything is wired right up to now) is to swap the front and rear connections at the head unit, i.e. connect the front outputs to the back speakers, pain in the ass, yes i know, but this will allow you to rule out whether the head unit itself has a problem. You could also get another pair of speakers to try it, stock skyline ones are not that great anyways so changing tem would be a good sound investment!