How would adding a score of 0 to this affect the mean and the median game scores

The median would be affected because there are more numbers and the "middle" number will change. (ex. 12345 vs. 012345)

Although then you find the mean, 0 will not add any value, but you still have to include it as a value when you divide by the number of numbers. (ex. 0,1,2,3,4. Added together they equal 10, but to find mean you must divide 10 by the number of numbers. 10/5=2)