A Gaussian noise is a random process which, when simulated, produces realizations added to the image. First, let us note that the image is of type
uint8, with integer values from $0$ to $255$. A random effect is often of
float type. Here, adding a noise to
uint8 data yields
uint8 data. Hence, you have an additional quantization effect (a rounding of non integer value). This effect is completed by a saturation effect: negative values are cast to zero, values above $255$ are set to $255$. So positive noise components plus saturation often means more brightness. Conversely, negative noise yields darkening.
Let us now look at the effect on a single pixel. Add a Gaussian noise with
average $\mu$ and variance $\sigma^2$. Since one realization takes values in $]-\infty,\infty[$, it might happen that the realization has a negative value. Hence, the noisy pixel will be darker. From the Gaussian empirical rule, you have about 16% chance for that with your parameters (chances that the realization is below $\mu -\sigma = 0$). For one given realization, it might happen that all the noises for each pixel are negative, around $0.16^N$ if $N$ is the number of pixels, and the image will be darker (up to quantization). So it is quite unlikely for a sufficiently large picture.
Does mean increases the brightness?
If it is positive, yes, with a large probability. Especially since the image is saturated
Does variance increases the amount of noise?
If you consider your image as clean, yes, in general. However, if your image is already noisy, adding an uncorrelated noise can, in some cases, counterbalance the
natural noise. In your case, i'd say yes. However, adding noise can increase the "quality", as used in dithering for instance:
Dither is an intentionally applied form of noise used to randomize
quantization error, preventing large-scale patterns such as color
banding in images.