I was expecting (or hoping) that the pictorial versions would somehow illuminate why these various identities hold. But they seem to be just an alternative set of symbols.

I wonder why numerals are used to represent 2 and 3 in some places, but stacked boxes are used in others?

The idea appears to be that artists find the use of colors easier to memorize / mentally manipulate compared to letters for variable identifiers. I'm not an artist, so I'm curious if others find this to be true.

When I imagine mathematical operations like these, I sometimes imagine numbers moving around, distorting, and fading in and out in a way representative of the calculation. For instance, “x^2 * x” would show the x on the right moving onto the x on the left, and the 2 fading down to be replaced by a 3 fading in, like an odometer. I’d like to some day create a Flash animation of some calculation showing those movements.

What is the one with the eyeballs? I don't get that one.

Looks just like variablekitten from Abstruse Goose: http://abstrusegoose.com/345