MAD and RMS SNR relation
The two main measures of scatter in the Gaussian case are the mean absolute deviation:

$$d_n = 1/n\sum |x_i-\overline{x}|$$

and the mean square deviation:

$$s_n = \sqrt{1/n\sum (x_i-\overline{x})^2}$$

If the random Gaussian noise has deviation $\sigma$, $s_n \to \sigma$ and   $d_n \to \sqrt{\frac{2}{\pi}}\sigma$. I don't have the exact references right now, you can find them in Robust Statistical Procedures, 1996, P. Huber, or texts by Hampel, or Rousseeuw, on robusts statistics.

The derivation is OK, let us use the zero-mean case: the expected value of the absolute value of the random variable: 

$$E ( | x | ) = \int_{- \infty}^{ \infty} | x | p ( x ) d x$$

then you derive:

$$E(|x|) = 
2\int_0^{\infty} \frac x {\sigma\sqrt{2\pi}} 
e^{-\,\frac {x^2} {2\sigma^2} } dx$$

then 

$$E(|x|) = 
\left[-\,\sigma \sqrt{\frac 2 {\pi}}  e^{-\,\frac {x^2} {2\sigma^2} }\right]_0^{\infty} 
=  \sigma \sqrt{\frac 2 {\pi}}$$

But this is for the mean absolute deviation, I am not sure of the equivalence with  the Median Absolute Deviation.