No. The binomial distribution (discrete) or uniform distribution (discrete or continuous) are symmetrical but they are not normal. There are others.
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No. There are many symmetrical distributions that are not Normal. A simple one is the uniform distribution:
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They are both continuous, symmetric distribution functions.
No. The Normal distribution is symmetric: skewness = 0.
The t-distribution is symmetric so the question is irrelevant.The t-distribution is symmetric so the question is irrelevant.The t-distribution is symmetric so the question is irrelevant.The t-distribution is symmetric so the question is irrelevant.
Yes. However, because the distribution is symmetric about 0, some tables give only positive values for z.