I want to use a transformer to both 'unbalance' the audio and to isolate it. Is this possible?
Yes.
All diagrams I've seen show transformers with a centre tapped primary used to unbalance audio. But that means I have to connect the centre tap to the ground of the left hand circuit.
Do I need the ground from the left hand circuit at all? Can't I just use a transformer with a single primary and secondary? That is, not have any reference to the ground of the left hand circuit at all, in the right hand circuit?
Because you want galvanic isolation, the center tap of the transformer should be connected to the ground of the transmitter, rather than the receiver. This means your cable should have a third conductor, a "signal ground" along with your differential twisted pair.
An alternative topology is to put the galvanic isolation on the transmitter side.
simulate this circuit – Schematic created using CircuitLab
For the receiver to be balanced (which is necessary for the twisted pair to be balanced) the values of Rhot and Rcold should be equal, and likewise, the values of Rf and Rg. The resistors should be 1% precision or better, for good "balance" and (common mode) noise rejection.
If the cable is not too long, (and the frequency is audio, as it is in this question), it is not necessary to match the receiver's impedances with the characteristic impedance of the cable. However, if the cable is long, (or the frequency is above audio, which it is not in this question), then the receiver's impedances should match the characteristic impedance of the cable. To do so, the resistor values should all have a value equal to 1/2 of the cable's characteristic impedance. For Cat-5 or Cat-6, this would be 50\$\Omega\$.
The twisted pair is balanced according to the definitions given in Wikipedia and this stack exchange answer. That is, the impedance from either line to ground is the same. By having the impedance to ground in both lines the same, common mode noise (such as "hum") that is picked up on the line can be rejected by the receiver's differential amplifier.
[The twisted pair is NOT "balanced" according to some (misguided) criteria. That is, it is not +6dB higher than the original signal. Nor can you double the voltage by shorting one of the lines to ground. This are commonly tests for "balance", but are, in fact unrelated to "balance"]
My second question is. I remember reading somewhere that connecting ground to chassis at both ends of a long run can cause noise. People suggest not connecting the cable shield to chassis at one end of the cable run. Is this nonsense?
No, it is NOT non-sense. The "people" you refer to are correct. Connecting a shield to ground at both ends creates a ground loop. Ground-loops can induce noise into your signal. Don't connect the shield to ground on two sides, just one.
