The math of gearing is pretty interesting, except I always seem to find it written in Red-Rubber-Ball-Table-Engineer-style, which tells the narrative of known results in almost exactly the opposite order that I, wearing my incorrigibly cranky Research Mathematician hat, want to hear: answer first, interesting properties of answer maybe second, explanation of what's really going on and why buried in a footnote if you're lucky.
If I reaaaally just wanted some involute gears, I could just run the black-box involute-gear plotting program, sure. But the anxiety of possibly in the future wanting to do something slightly different than the stock answer drives me absolutely batty. I wanna know how it works, I wanna know why the involute curve is the unique curve satisfying desirable properties X and Y and Z, I wanna generalize it, maaaan.
Anyway I have reconstructed a ~50-75% satisfying answer to most of the particular questions that were bothering me. The "line of action" arises pretty naturally from some calculus and demanding that gears only slip and do not interpenetrate while undergoing constant rotational rate. If you assume the rack of a rack-and-pinion has trapezoidal teeth (which makes sense, since a rack is an infinitely-big gear, and teeth seem to get flatter the bigger the gear) then an involute curve is exactly what consistently engages a rack-tooth along the line of action as it moves. I'm missing only understanding why the addendum height is what it is, but I assume that some visualization might help me guess exactly what thingy it's maximizing.