First, you have to understand why there are two tests, for a same quantity. Let's say you have a sample $x_1, \dots, x_n$, drawn from an unknown distribution and you want to test if the mean of the distribution is zero or not.
So you compute the sample mean $\overline x = {1\over n} \sum_{i=1}^n x_i$. And you compute the sample variance $s^2 = {1\over n-1} \sum_{i=1}^n (x_i-\overline x)^2$. And finally, you reduce $\overline x$ by the standard error $s = \sqrt{s^2}$, considering ${\overline x \over s/\sqrt n}$.
There are two cases :
the underlying distribution is normal ; then ${\overline x \over s/\sqrt n}$ is distributed like a $t$ distribution (if the mean is zero), and you use a $t$ test. This is an exact procedure.
you don’t know whether the underlying distribution is normal or not. If $n$ is big enough, the central limit theorem tells you that ${\overline x \over s/\sqrt n}$ is approximately distributed like a standard normal distribution (if the mean is zero), and you use a $z$ test. This is an approximate procedure.
What you were stating are just guidelines to help you decide if the assumptions required for $t$ test are satisfied.
I don’t get rule 3. For me, it is just false. If the distribution is skewed, it is not normal, and you have no reason to think that the $t$ test will perform better than the $z$ test.