Good. As you revealed more and more detail, I was ever more convinced that this was nothing to do with the cabling running parallel and close proximity to mains cabling. And I was pretty convinced from the start !
Once upon a time, a hum on a line was usually put down to earth leakage on one of the pair. I.e. loss of balance. People could live with that hum; but the internet is no so flexible. BT's response to this was the "Iplate", a device sandwiched between the Master Socket and your house's telecom winding. This was advertised as a line re-balancer - which it was - but it also filtered the bell wire. If you have one it may be worth trying it - either at the master or the Ext2 socket.
As for why, I assume that there is some electronics between what should be a balanced system and your ears - maybe in that little bulge on the headset wiring. I'd guess that some of that circuitry involves some pretty high impedances - making it more susceptible to fields around it. As you say, making it work like an aerial.
Just to give an example of this, this PC is connected by twin, screened cable to my hifi. It works fine - until I pull the jack out of the PC, when I get a loud hum. Why - when it is, after all, just a couple of metres of shielded wires?
The amplifier has a relatively high input impedance, so only a little induced hum current manifests as quite a high voltage (Ohms Law). Generally PC LINE OUT sockets expect to see about a 32 ohm load. I could totally fix this by connecting a 33 ohm resistor between each of the pairs' screen and inner. Just too damn lazy, I guess !
As for landlines, we once had three, now just one. The number we give third parties is my mobile number, so I guess that we are part of the transition. It took a long time for me (professional engineer) to convince my wife that the broadband didn't need its own landline - and then to convince her that we should integrate landline and internet with one ISP to radically reduce costs.
We've saved several thousands, since - plus there's the benefit having the phone available (almost) wherever we are.
Gotta go, three texts already this morning warning me that I'm overdrawn !!!
PA