What is happening in this basic push-button circuit?
what would the equation be to identify the amperage going into pin 2 when the button is pressed? 


Once the button has been pressed for more than a few nanoseconds the current will be 0 (or as near as).

Immediately the button is pressed the formula is  I(t) = dQ(t) / dt since the input pin is a MOSFET gate and as such is seen (in simple terms) as a capacitor.


  What if I used the 3.3v on the arduino or a 9v battery instead?


The same.


  What is happening to the input pin when the button is not being pressed? (I assume it's a guarangeed LOW state due to any current being sapped by the GND, yes?)


It is pulled down to ground by the resistor.


  What would happen without the pull down resistor? A short when the button is pressed? What would happen without the ground being connected at all? Anything?


The input will be HIGH when the button is pressed, and floating (initially HIGH) when released.


  A tutorial for how to choose a pull down resistor I seen suggested a resistance value of 10 times the impedance of the pin. I'm not quite sure where to find this impedance value, but I have seen on an arduino tutorial "A 10K resistor is a good value for a pullup or pulldown resistor." Just not much of an explanation of why that value.


The main concern with a pullup or pulldown resistor is the current it demands from the power supply when the button is pressed. If it's too low then you will be wasting current. If it's too high then the input may take a while to turn off when you release the button.  10K is a "good" value since it is a good balance between those two factors.

The current from the supply is calculated as I=V/R, so 5/10000 = 0.5mA.  Low power systems may choose a higher value (100kΩ, say) to reduce the current.