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Do normal probability distributions have different arithmetic means and different standard deviations?
Yes. And that is true of most probability distributions.
What is the benefit of transforming standard normal distributions to conform to the standard distribution?
There are no benefits in doing something that cannot be done. The standard normal distribution is not transformed to the standard distribution because the latter does not exist.
Do some normal probability distributions have different means and different standard deviations?
Yes. Normal (or Gaussian) distribution are parametric distributions and they are defined by two parameters: the mean and the variance (square of standard deviation). Each pair of these parameters gives rise to a different normal distribution. However, they can all be "re-parametrised" to the standard normal distribution using z-transformations. The standard normal distribution has mean 0 and variance 1.
Do some normal probability distributions have different arithmetic means and different standard deviations?
Yes. Most do.
Why normality is required for standard deviation application?
Because the z-score table, which is heavily related to standard deviation, is only applicable to normal distributions.
