Ok, trying to answer the questions and suggestions in sequence:
A. the Processor, with it's normal 'read VCC' function, will read the supply voltage to the Processor. Not the supply voltage to the voltage regulator.
B. Yes, the voltage regulator consumes some quiescent current, but it is very small compared to your overall budget and supply (AA batteries). Mentioning coin cells in this context is probably more distraction than helpful.
C. Yes, you can measure other voltages, including the battery's, if you use an analog input, a voltage divider and scale the reading by the measured VCC but there are two factors you need to consider before doing this:
1. The resistor divider will draw cpointontinuous current from the voltage source - generally not a good thing in a battery operated device.
2. The measurement usually uses VDD as the reference (to get good scaling), but if VDD is variable (and it ALWAYS is - to some extent) then you might need to measure VDD using the bandgap reference and then scale all other measurements accordingly. Now one could ask that if you can use the bandgap to get the basis for VDD, why not use it for all other measurements? Mainly 'cause its easier to use the standard reference (VDD), but since it's your processor, you can do anything you want
Also, if all you care about is whether the battery is doing well or is getting weak, measuring against the 3.3V VDD is probably accurate enough.
3. (extra bonus factor): You need to learn about the input resistance vs capacitance of analog inputs in order to get a reliable reading from a high resistance source. Plenty of discussion of this here and on the web.
Net: If you absolutely NEED to know the EXACT battery voltage and it is greater than VDD, you must use a voltage divider and I recommend using a load switch to switch the divider off when not reading. If you just need to know whether your battery is dying, then reading the VR output MAY be sufficient. However, this is only generally true if the battery still has some 'headroom' at <3.3V (this rules out most batteries except alkaline and possibly SLA).
Tom