Yes, the location tracking (wether that be simply asking, or IP checks, etc) is mostly for regulatory compliance.

Salaries adjusted by region is pretty common. My employer does it. My wife's employer does it. Etc.

I don't know how the US works, but in Canada I've had jobs where the employer took no taxes off and I had to pay them myself at the end of the year. I loved it, I can understand why many can't handle it though but the increase in cash flow actually increases your investment opportunities short term.

If this is true for the US too then Facebook is full of shit and using this as a weak argument for why they have to know if you snuck out of the city to go live in a cheaper area.