1. Can a normal distribution not be symmetrical?

Few distributions are perfectly symmetrical. Often a distribution will be skewed,which means it "leans" one way. Kurtosis is a term associated how "pointed" or how steep a normal curve is.

In many cases, the larger the sample, the nearer the curve comes to being 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.