Can I use an image I download over the internet instead of photographing the ColorChecker myself?

No. You must run the script on an image of the ColorChecker captured with your camera in order for the results to be of any use to you.

Do I need to calibrate for every possible lighting contition I will shoot in?

No. If you calibrate in daylight or 5000K conditions, it will work for most of the lighting situations you encounter. The only special case is tungsten light (3200K) which contains significanly less energy in the blue portion of the spectrum. If you shoot with tungsten light, you will likely benefit from performing a calibration under those conditions.