It is symmetrical
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Yes, it is true; and the 2 quantities that describe a normal distribution are mean and standard deviation.
No. The Normal distribution is symmetric: skewness = 0.
The standard normal distribution is a special case of the normal distribution. The standard normal has mean 0 and variance 1.
The standard deviation (SD) is a measure of spread so small sd = small spread. So the above is true for any distribution, not just the Normal.
The domain of the normal distribution is infinite.