Amount of information in scaled-down images Does an image that is scaled down lose more Information when 
calculating averages of pixels rather than selecting single pixels?

One way to scale down an image is to replace 2x2 pixel blocks with the average of those four pixels, the other way is to just select the top-left pixel (or any other) instead.

Averaging is done by summation and integer division. We reduce the Problem to just scaling 2x2 images down to 1x1 images. For black and white images there are 16 possible images, 15 of those will be black and one will be white
thus the entropy of the scaled down image is 0.337 bits. For grayscale images with 5bits per pixel results
in 4.2524 bits, with 6bits per pixel we end up at 5.251 bits. 

Does averaging really lose more information than just selecting top-left pixels?

I'm aware that you lose some information due to rounding but my calculations don't seem to converge so rounding wouldn't explain everything.
Averaging-and-truncation certainly loses more information. It is not because you are not preserving any specific pixel: it is because you are taking an even probability distribution and giving it a strong bias in one direction.

If I have understood you correctly, you have taken the 16 possible 2x2 images and transformed one of them to white and 15 of them to black. This 15:1 split is about as bad as you can get! It destroys information, and it is the reason why your results are so bad compared to the "take top left corner".

The nearer you get to an "8:8" split, the less information you will lose. So try averaging and rounding: what I think you call "taking the median". This will split 11:5, which is less bad. It is still not very good, because the 6 cases of "2 black, 2 white pixels" all have to be translated either to black or to white - that is why you can only get 11:5 rather than the ideal 8:8 between black and white outcomes. [It would be interesting to try "use the majority value if there is a majority value; if not, use the top left"].

If you try $3\times 3$ blocks then either a clear majority of pixels will be white or a clear majority of pixels will be black: a straight 256:256 split. This is the best possible outcome, and if you simulate it then you should find it leads to the least information loss.