In StyleGANs the random noise goes through fully connected layers at the input to generate w vector.
Does it mean that during back-propagation neural network for noise also gets trained?
Second question is why do we in general use a Gaussian noise? Can we take any other probability distribution say uniform for example?
Yes, the training also applies to the part of the network that takes the noise as input. How that behaves affects the results, right? You start with random weights, so random weights applied to random noise will likely give pretty random results, when you first start the training. That probably won’t be your final goal. 
As to the question of why Gaussian, I don’t think there is a definitive answer. This is an experimental science: you can try using different distributions and see how that affects the results. But unless your application requires the noise to look like a probability distribution, Gaussian is a plausible first choice because it models the behavior of natural phenomena. Try it as the first solution and then if the results don’t meet your requirements, you can try other distributions.