For the earphones where the voltage is relatively high and he mpedance relatively low, the chances of interference strong enough to break into the wiires and be audible is very slim. The microphone is quite different though, it will have high impedance and the voltage will be only a few mV. Almost certainly, whatever you are plugged into will have a microphone amplifier in it which not only amplifies the microphone signal but everything picked up on the cable as well.
This is the reason why screening is used on microphone signals, it gives a better ground connection and by physically surrounding the inner signal wire, it shields it from outside influences. Certainly, if you have individually screened pairs in the network cable it will help but whether it screens well enough for audio is debatable, you would have to try it to see. Don't forget that network cable isn't designed for continuous flexing and you might find it either breaks easily or causes electrical noise as its moved because of charges building up on the plastic insulation around the wires.
Brian.