1. Is the square shape of the image that obm presented caused by the "averaging of the compression features of jpg? With a bright circular object averaged with dark space around it?
Do you mean the square or rectangular shape of the "windows" around each object in that image?
If so, no. The picture is broken up into 8x8 pixel blocks, and each block is encoded separately from all other blocks. A block is encoded as a (relatively) high precision DC value, which gives the average brightness of that block, plus a bunch of AC components that, when combined, say how much each pixel in the block differs from the average. These values are computed with a relative of the Fourier transform called the (2D) cosine transform. In essence, it breaks the image down to a weighted sum of a whole bunch of checkerboard patterns, both square and rectangular. The weights are the AC components.
There's as much data in these components as in the original image, so we still haven't compressed it. But we can
quantize the AC components, i.e., reduce their accuracy, and if we haven't done that too aggressively then when they're turned back into pixels they won't look much different from the originals.
Because pictures often contain flat areas without a lot of fine detail, it is often possible to squash most or all of the AC components in a 8x8 block to zeroes. These are especially easy to encode using "run length encoding", i.e., you simply say how many consecutive zeroes there are without actually sending them.
Because Apollo pictures often include lots of flat black sky, they're ideally suited to JPEG compression (look at how small many of the .jpg files can be).
A similar effect occurs in voice communication systems, both analog and digital, where very weak sounds that you can't hear anyway are removed to avoid wasting system capacity on them. (Analog systems have VOX and squelch; digital vocoders quantize much like video, sometimes with variable data rates.) So very dark areas of pictures often get squashed into pure black, removing even the film grain, and voice pauses with only background noise get squashed into pure silence.
This obviously doesn't happen in the non-black parts of the picture, so the combination of quantization errors and film grain turn into random-appearing noise when the picture is reconstructed. This includes originally black pixels on the edge of the block that are adjacent to completely black blocks. That's what makes the edges of those blocks so conspicuous. But that only happens when you fiddle with the brightness to make them visible. And that's unfair to the compression algorithm, because it's not designed for that. If you don't want any compression artifacts at all, don't compress.
