Completely fair? They're getting paid to work, not to live in expensive houses.

The relevant issue isn't "fairness", it's "market value". There is a lower supply of people willing to work in regions with high cost of living (for the same wage).

If I'm working remotely, I'll go to whomever pays the highest (hint: it's not the company that gives me a salary cut based on location).

If.

I am working remotely. I'm getting paid "London" rates, not "Greek village" rates.

you're right, it's not about it being fair, it's about it being the market rate for your skills in your area. If you want to live in Podunk, Nowhere, how can you expect an employer to pay you twice what anyone else in your town is willing to offer you? Especially considering that you'll be mostly telecommuting. Most companies prefer to have you working in their offices. Paying NYC rates to someone working from home, in a low wage area, makes no sense.

What should the company be bothered about what and where I'm spending my money. If I spend more will my work become more valuable to the company or vice versa?

On the similar lines I don't understand the concept of companies asking people about their compensation in the previous company. You are supposed to pay for my work, how does my compensation even matter here?

If my previous compensation was high will my work in the current company become more valuable or earn higher profits?