1. Can a normal distribution not be symmetrical?
Few distributions would be perfectly symmetrical. Often a distribution will be skewed, which means it "leans" one way.
In many cases, the larger the sample the nearer the curve comes to being a normal shape.
2. Are all distributions normal?
No! There are rectangular distributions, bi-modal distributions, triangular distributions...
2. Why are there no tables for negative values of z?
Because of the symmetrical nature of the normal curve the probabilities associated with the left hand side of the curve are the same so they are not repeated.