the rate of noise presented in images
You could check out my MATLAB function, which estimates the variance of the noise. It is also mentioned in a Mathworks blog post, in comparison with other methods.

Regarding $\sigma$, you mention in the question, simple answer is yes, longer answer is no. It refers to the variance of the noise normalized with image dimensions as follows:

$$\sigma = \frac{\sigma_0\sqrt{0.5\pi}}{6(W-2)(H-2)}$$

$-2$ is due to the convolution kernel, applied previously. However, this is not a final quantity, giving you the absolute noise level. Nevertheless, for the variance, you could use it as a percentage, just like you mentioned.