Hi Zach,
1) I'm glad that you are not using John's 0.1uF cap across the input, because when your 10K volume control is half-resistance, it will cut all frequencies above only 640Hz. Maybe that was to filter his buzzy speech synthesizer.
2) See if the TDA7000 is able to drive a 10K volume control. Harry's 100nF output cap will rolloff low frequencies below 160Hz when driving a 10K volume control. Maybe a 20K volume control is more suitable.
3) You must use a ceramic supply decoupling cap of 0.1uF. It should be mounted as close to the LM386's pins 6 and 4 as possible. The 0.047uF (not 0.0047uF) cap at the output should also be ceramic. You will also need a 100uF cap mounted anywhere and connected across the battery input to all circuits.
4) You don't need a ground plane for this simple audio amp. An ordinary PCB or even Veroboard will work fine.
5) You don't need a bypass cap at pin 7.
6) Don't use a resistor in series with the 10uF cap across pins 1 and 8. It will add extra gain only at low frequencies (bass boost) which you certainly don't need since your receiver has de-emphasis (like all FM receivers have), but your simple transmitter doesn't have a corresponding pre-emphasis (like all FM radio stations have) so it will sound very muffled and bass-boost will just make it worse.
7) Your headphones will need about 500mV RMS for loud peaks. Harry says that the TDA7000 produces 70mV, therefore you need a gain of at least 7.1. The gain of 20 without the cap across pins 1 and 8 will be fine (enough extra gain to turn-up the volume when the input audio is low).
8) Use a logarithmic, not linear, volume control.