Wiring a Logitech extreme 3D pro joystick for Xbox adaptive controller
Yes, you can use a 2 channel (L,R,GND) as a mono-cable. You can wire both R&amp;GND together and use is as a monocable. Like here in the picture (left and middle connector).



(Image source: Cable Chick - Understanding TRRS and Audio Jacks)

Why the diode-checker beeps when you connect the button to the device ?
The diode checker checks for voltage (it provides also itself). The device you connected it to will supply some voltage to the button ... even in the open-state. So that is totally normal. Your measurement device picks up that voltage and beeps.

10 Buttons are a lot of cables. I don´t think you can decrease the thickness of the cables too much without grinding the isolation a bit. You could use a 2 ethernet cables (each provides 8 lines to use) to make a longer cable run from the joystick to the adaptive controller. You could also use some cable-zip-ties to hold them all together or these handy velcro-ties:



(Image source: Show Me Cables)

Seeing the circuit board with the opened traces on the PCB, I think you are on a great start. Please don´t see this as a complete answer, just some hints.