VSD with Power Cable Only

Maybe, but I would guess that it would be more economical and trouble-shooting-friendly to provide separate control wiring. (That's just an estimation.)

BS7671 requires the emergency stop button to be a separate hard-wired circuit.

Probably but it will cost way more than what running a control cable will and as said the E-stop at the remote panel has to be hard wired anyway.

Pull the cables and stop trying to be so cheap. You're gonna hurt somebody if you don't.

There is a technology called a Power Line Communication Carrier (PLCC) that can do it, I have used these on overhead crane controls. They are not a discrete or analog signal though, they would be digital (serial) , so you would need something interpolating the information at one end at least, maybe both depending on the drive. But as mentioned, you cannot get these to do Emergency Stop functions. Also, there are numerous pitfalls to watch out for with using them on circuits with VFDs. Not all VFDs are created equal.

that depends on what you mean by "control"

In the oil patch in particular in the oil wells, we use VSD's to power the downhole pumps (ESP's) and at the bottom of the motor we fix a downhole monitoring gauge that will relay to the surface, downhole pressure, pump discharge pressure, downhole temperature, motor temperature and motor vibration. This data is transmitted to the surface via the power cable connected to the motor and extracted at the surface off the the power cable.

The gauge surface equipment extracts the data and it will display it at surface and/or transmit it further to a control room, as well as being able to control the output of the VSD if required to maintain a certain flow rate/well head pressure.