No. Normal distribution is a special case of distribution.
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It is not necessary that all symetric distribution may be normal.
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No. The binomial distribution (discrete) or uniform distribution (discrete or continuous) are symmetrical but they are not normal. There are others.
No. The Normal distribution is symmetric: skewness = 0.
If the two distributions can be assumed to follow Gaussian (Normal) distributions then Fisher's F-test is the most powerful test. If the data are at least ordinal, then you can use the Kolmogorov-Smirnov two-sample test.