I also get some noise from the Darlington. Sounds like a Square Wave. Frequency is loosely related to the guitar pitch. It kind of motorboats on release. Disconnect the Base, no noise.
Sounds like your Darlington might turning on/off too fast (needs filtering), and with too drastic of a transition (too much gain). This might be dumping too much current onto the ground rail. That current seems inherent to the design, though, so it might be an artifact of the breadboard and the high-ish resistances it involves. A build with a better ground connection might not have the same problem, though it's probably wise to solve it on the breadboard anyway.
I would try removing Q3, replacing it with a series diode and shunt cap(?) to create an R-C filter. Using a diode in place of the resistor will keep the compression threshold the same as using two transistors. Using a single NPN instead of a Darlington pair may reduce the gain of the LED driver and maybe not turn it on/off so hard.
The quick version of this experiment would be to just lift Q3 collector and let the base-emitter junction be your test diode.
Hopefully you don't need a filter, but a shunt cap to ground between the diode and Q4 base will help slow down transitions if you need to. I have no idea what value to use.
I also find it interesting that the Negative Feedback is going from Q2 Emitter to Q1 Collector.
I guess since it's two inverting stages, a connection back to the Q1 Base would have been Positive Feedback.
Q2 emitter feeds back to Q1 base, just like a Fuzz Face.
Q1 emitter feeds forward to Q2 collector. Basically when LDR is low-resistance, Q1E dominates the output, otherwise Q2C dominates the output.
This whole thing is basically an auto-panner between an input buffer and a cleaned-up fuzz face, controlled by the LED/LDR, which is pretty cool.