What is the appropriate measure of average that must be used?
The answer depends on the distribution of the data - in particular, its skewness and the presence of outliers.
How is the choice made between mean and median to describe the typical value related to the shape of the data distribution?
A z-value by itself, has nothing to do with level of confidence. A z-value can be used to calculate probabilities of observing a result that is at least as far from the mean. That probability measure can be used to calculate the level of confidence but you need to be careful about using the one-tailed or two-tailed measures - as appropriate.
What are the metrics that can be used to measure the efficiency or effectiveness of Incident Management?
Occasionally, estimation must be used to account for objects, like body cells or other very small things. In measurement, techincally, everything is an estimation -- unless it is an integer. How long is the block? Do you measure it to the neaest inch, tenth inch, thousandths of an inch? Reminds me of the question: How long is the coast line of England??