Are 2nd Order Edge Detectors More Susceptible to Noise?
If you assume the Edge Detection is SNR driven operation, one could find a Mathematical justification for this.

First, the variance of Additive White Noise with Variance $ {\sigma}_{n}^{2} $ at the output of a Linear System given by $ g $ is $ {\sigma}_{n}^{2} {\left\| g \right\|}_{2}^{2} $.

Let's look on an Image filtered by a derivative approximating filter $ g = \left[ -1, 1 \right] $.
This increase the Noise Variance by a factor of 2.
For 2nd order the filter is $ {g}_{2} = g \ast g = \left[ 1, -2, 1 \right] $ hence the factor is 6 (It comes from the reason the second time we filter the noise is correlated).

Now, the question does the signal (Edge) is amplified by the same factor?
Moreover, on areas where there is no edge, the probability to detect noise as edge increases.
To see intuitively, just operate 1st order method on the gradient of the image and see how much False edges you have now.