Why should an image be blurred using a Gaussian Kernel before downsampling?
An image &quot;should not be blurred using a Gaussian Kernel&quot; in general. This can be a safe bet for a lot of basic image processing needs, and a smoothing is almost mandatory when you want to control the information lost by the downsampling.
Blurring is (often, not always) another word for low-pass filtering. When an image contains high-frequency content (fast variations), downsampling without blurring can produce visually weird, annoying aliasing artifacts or moiré patterns. There is an example on wikipedia Aliasing: the original

and the downsampled, represented at the same size:

The ripples on the bottom right are low-frequency artifacts generated by a careless brute-force downsampling. A blurring would attenuate image sharpness, dim the borders between bricks, and reduce the apparent aliasing aspects.
The choice of appropriate blurring filters has a long history in image processing. Gaussian shapes have long been considered somehow optimal for different reasons for &quot;theoretical&quot; continuous images. Plus, it is both decreasing and symmetric in  the space and the frequency domains. In the time domain, this means that faraway pixels have less influence. In the frequency domain, frequencies are reduced monotonously from low to high.
Since most images are discretized, reality is somewhat different. Since the Gaussian convolution  used to be computationally expensive, early approximating filters were designed with short support, borrowed for instance from Pascal triangle. Later, fast recursive implementations were designed (Deriche, Shen, etc.)
I guess question 1) is answered. For question 2) simple averaging gives an equal weight to all pixels in the window. Hence, faraway pixels are given equal importance with respect to closer pixels, which is not optimal in regions where images exhibit weak stationarity, like trends, edges and textures.