If you draw 21mA from the 6V regulator output the regulator will dissipate 6V * 21mA = 126mW as heat.
Ptotal = Iload *Vload + Ireg*Vreg
The battery will still only run for 1hour theoretically. I say theoretically because the capacity is usually rate at a lower discharge rate.
that is correct for many regulators oftain referred to as "buck" regulators.
a "switching" regulator on the other hand pulses the power (kind of like PWM) and then uses capacitors to give an average voltage.
this kind of regulator does attempt to give twice the battery life at half the voltage
but they are never 100% efficient.
80-90% efficiency is typical with switching regulators.
they usually require a few more components and a little more expensive but not much.
alternatively, Dimension Engineering (one of SORs advertisers) produce a switching regulator module that has all the extra components included in a 3 pin package.