From what I understand though, these formulas depend on the probability law for y being a normal distribution.
This is not true: In the first three formulas, the results follow from properties of variance that apply regardless of distribution. (And the fact that $a,b$ are known constants.) To practically demonstrate this, try generating random samples from any distribution, then calculate the sample variance or standard deviation. Compare this with the variance or standard deviation given in the table, and you'll see that they match:
> x1 = runif(100)
> var(x1) * 4
The others appear to follow the Taylor series approach detailed above the table. (I verified this for $f = a \log(bA)$, but leave the rest to you.) As noted in the text above the table, how well this works depends not on the distribution, but on how well a linear approximation fits the function:
Error estimates for non-linear functions are biased on account of using a truncated series expansion. The extent of this bias depends on the nature of the function. For example, the bias on the error calculated for log x increases as x increases since the expansion to 1+x is a good approximation only when x is small.
In other words, you may use these formulas with the knowledge that error is not normal; but, how well they work depends on how well the function can be linearly approximated. Other methods of error propagation are found here, linked from the same article.