Wire choice for ground pin

There is no "ground pin". Voltage is a difference of potential between 2 points - so as you would say "signal" and "ground" are the same.

Do you mean a single strand from a cat-5 or cat-6 cable? If so the rest applies.

Maybe, but not because there is anything particularly special about cryo treated lamp cord.

If you think of the science behind a cable: By using a thinner conductor for ground, you are increasing the chances that any currents induced by a ground loop flow in the signal wire. AKA, you are begging for a ground loop. If your system is not mains powered it dosnt matter, but maybe it does.

Regardless of the reasons that a thinner ground wire than signal is bad news in an interconnect you should use ALO (isnt it jenna wire?) wire for ground because it looks pretty and that accounts for more than 99% of building cables.

You should try different things and find out for yourself.

You'll probably get all sorts of different opinions, so I'll give you mine. I'm not good at stirring up arguments, no matter how controversial I get (probably because I scare off most people and the others just politely correct my <<ahem>> gestalt approach) but here's hoping...

In typical consumer equipment, the ground wire of an interconnect connects the DC grounds of the two pieces of equipment, as well as being a conduit for the signal. If there's the slightest difference in ground potential, a DC current will flow along the ground wire. The question here is whether to ignore this or not. A fat wire with low resistance will have higher current flow and could remove some of the potential difference, while a higher resistance will have a lower current but retain the higher DC difference. It's not clear what to do about this, or even whether to try. The obvious answer is to use fat wires, but a higher resistance is arguably a good thing here because you don't want to eliminate the difference this way. You'd only end up with the components fighting to get back to what they consider normal, and end up changing the currents internal to the components. Not that a wire's resistance will make much difference anyway.

The signal wire is often isolated from these DC differences. Some use a capacitor on the output to essentially block the DC. Note that this blocks the DC potential that appears on the output, which is not necessarily the 0VDC circuit ground. Some use a servo or other circuit to keep the output near ground potential. Either way, this is the same ground potential that might disagree with the other component.

Another (admittedly odd) way to look at things is that the electrical characteristics are different for the two wires. The signal wire sees a very high input impedance, while the ground wire provides a low impedance return path. I'll get a lot of argument here, but I think this also affects the ac characteristics. The input circuit sits between these two wires. It seems a little simplistic to ignore the effects of the input circuit. The usual way of looking at the ac side is to assume that the high input impedance seen by the signal wire totally swamps any other effects, so everything else is ignored. The low impedance of the return wire also assures that the input circuit is in control of all the relevant electrical parameters on the return side. This again points to using a fat, low impedance wire for the return, and maybe also argues that it's not as important for the signal wire. Note that the case for headphone and speakers cables is different. The argument to this would be that ac isn't directional, which isn't quite true. Once you get into higher frequencies, you don't even need a return path. A capacitor provides the proof that this same principle holds for even very low frequencies.

Okay, this isn't so much an answer to your question as a bunch of other questions surrounding the issue. Maybe it's a justification for trying different wires on the two sides. It would take weeks or years to work through this in a more coherent way, and I probably don't have the background to do a good job, so apologies if the above isn't all tied together coherently. A general answer isn't really useful anyway because so much of this depends on the equipment you're connecting.

The bottom line is the same as the top line. You should try different things and find out for yourself.

The signal passes through the ground wire and signal wires. If you could hear a difference with the ALO cable for the signal lines, you'll probably hear an improvement with an ALO ground wire too. You might also want to consider that all the current from the signal wires goes through the ground wire so the ground wire could carry up to twice as much current as either signal wire.