In a similar vein, the weather icons are 800x800px images displayed at no larger than 20x20px

Those large icons are rather compression friendly compared to the dithered images. It would still be a relatively huge save at 20x20 (after resize to 20x20 with imagemagick and pngcrush -brute, the 5.5k clear-day.png goes down to 559 bytes).

800x800 indeed is excessive. We'll resize them to something smaller. Jquery is already on its way out. Thanks for the feedback!