Think of it this way. 3 gallons of boiling water has a set amount of energy. Depending on the required outlet temperature, you will need a certain amount of "cooling". This is accomplished by the amount of ice/snow you use in the cooler. You will need more ice/snow the lower the desired outlet temp.
But as was stated above, the actual outlet temp in your setup is controlled by the velocity in the chiller. If you run the liquid too fast, you will come out hotter than desired; too slow and the outlet temp could be too low.
And to complicate things even more, as you run your wort through the chiller, the contents of your chiller will rise in temperature meaning the time required to be in the chiller to achieve the goal temp will increase as more wort goes through it. To combat this, adding more ice/snow than what is mathematically required to maintain as consistent a temp as possible is desired.
So, in the end, if you are using ice/snow, your internal temps are approximately 32°F. Always add water since liquid transfers heat better than loose ice/snow. Add ice as your wort is going through as needed to maintain a 32°F temp. Control the velocity of the wort through the chiller with a valve to achieve the desired outlet temp. I would probably go colder than desired and allow the contents to naturally rise up to the goal temp.
And lastly, I would suggest adding a valve on the outlet of the chiller before the carboy and use this valve to control wort velocity. By controlling the flow at this location, the tubing inside the chiller will always be "full" and pretty much void of air. If you control the flow before the inlet to the chiller, then you have a chance of having half full (or some percentage less than 100%) tubing and this will require slower velocity to get the proper heat transfer.
