What does "the depth of the 13.0 Å feature differs by 7 sigma from zero" mean?
In this case itseems to mean that the depth of the line is 7 times its error bar below the continuum level.

Impossible to answer. You say it can't be done, but the authors say that they fitted a Gaussian.

You either use a rough estimate (attributable to Cayrel de Strobel 1988) of
$$\Delta {\rm EW} \sim 1.5\frac{\sqrt{RP}}{{\rm SNR}},$$
where $R$ is the resolution element (the FWHM) of your instrumental wavelength response, $P$ is the pixel size (both in the same wavelength units) and SNR is the signal-to-noise ratio per pixel. Or you use an integration method and combine the SNR of each pixel used to find the uncertainty in the integral. Or you fit a Gaussian but using the area under the Gaussian as one of the fitting parameters.