Gaussian pyramid: why needs the image to be downsampled
In the context of a gaussian pyramid, why is the image downsampled separately although the numbers of pixels are decreased through smoothing already? 


The Gaussian filter adjusts the bandwidth of the content of the image. The downsampling adjusts the spatial resolution of the image.

The Gaussian filter is a low pass filter. After its application to an image, only the frequencies within its pass band will be present in the resulting image. 

An image of dimensions $M \times N$ pixels can depict $\frac{M}{2} \times \frac{N}{2}$ lines per pixel of spatial frequencies.

If you apply a low pass filter at $\frac{M}{4}, \frac{N}{4}$, you have reduced the spatial frequency requirements for that signal by half. Where before you needed $\frac{M}{2} \times \frac{N}{2}$ lines per pixel, now you know that spatial frequencies in your image do not exceed $\frac{M}{4}, \frac{N}{4}$ lines per pixel. The rest of the bandwidth that an image of that resolution can support goes unused.

At that point, you can reduce the image resolution to accommodate the "new" limits of the bandwidth without any loss.

From another point of view, you use the low pass filter to ensure that the resulting image will not contain any spatial frequency that could be aliased if it was to be plainly subsampled (because ultimately, resizing an image means that it will have to be resampled and "fit" into a smaller space. This smaller space means less bandwidth). Once you ensure that no frequency will be aliased, then it is safe to resize.


  An image with 8 x 5 pixels smoothed with a gaussian filter of 3 x 2 pixels will create a 6 x 4 pixel image if the filter can only be slided within the boundaries of the original image without crossing the edges.


This is to do with choosing only the "valid" values in the output of the filter. Values for which all the weights of the two dimensional impulse response are applied to image pixels.

Hope this helps.