Do I need to screen audio and microphone signals from each other?
Microphone signals are tiny, so small amounts of noise can be big relative to the actual sounds signal you want.  Microphone wires should be shielded, with nothing else inside the shield.

Signals going to a headphone are much lower impedance and much higher voltage.  Noise that totally overwhelms the audio signal coming out of a microphone might not even be audible on top of a headphone signal.

If in doubt, run separate shielded cables for the microphone and the headphone signals.  You definitely don't want them together inside the same shield.  The strong headphone signals can then cause feedback into the sensitive microphone lines.  This can lead to whistles and other bad effects.

You may be able to get away with twisted pair for the headphone signals instead of a shielded cable, depending on how strong the RF is and how the headphone amplifier reacts to RF being fed back into its output.