If touching the ground does anything, that's an indication that your grounding is incorrect. The only thing to do then is to figure out what you did wrong and fix it.
In your case, things are a little more complicated because you mixed modules from different manufacturers. But the first thing to try is to run just one ground wire from each module and have them all connect together in one place. You don't want your modules connected together through more than one ground path. Just connect the hot wire of each signal from one module to the next. The problem with jacks where the ground touches the chassis is that, although the chassis is ground, it allows multiple paths...
1) To mix left and right to mono, you use the same scheme as mixing different inputs: You start with a low impedance source (output of an amp stage), one for left and one for right. You connect each one of those through a large resistance then the other side of the large resistors you tie together and connect into your headphone amp input (high impedance). This allows the left and right to mix into one channel with relatively little loss.. but very little signal leaks between the left and the right bus.
Every circuit has a little bit of noise, if you want a mixer with zero noise, you have to leave ALL the parts out.. but that wouldn't work very well as a mixer.
If you design the circuit correctly there will be very little noise and it won't be a problem. After all, the professionally built mixers have hundreds or even thousands of resistors in them and they work fine with very little noise! If you use the resistors correctly you won't have a noise problem.
And you will not lose that much signal.. because.. let's do the math. Let's say the input impedance of your amp stage is 100K, and you use a 220K resistors to mix your signals.. you will lose 10db, if you use 100K resistors, you will lose 6db.. which you can easily make up in the amp stage.
But the point is, if you want your mixer to work, that is how you do it.
2) Phone lines (analog ones) are balanced! That's why phone signals can travel for miles with very little noise.
The first "audio" systems were phone systems, and many of the ideas used in pro audio came from phone systems.. including the signal levels we use, balanced lines, amplifiers, microphones, the decibel.. were all invented for telephones.. NOT for pro audio, which came along much later.
So yea of course phone systems don't need wide frequency response or high power.. but the principles are the same.
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