How do you get repeatable results each time the document is rendered, and if you adjust the various star parameters, are the positions adjusted accordingly without being re-randomized?
Glad you asked :) That's the trick :)
Each point (x,y) gets its own unique 32-bit int (well, not truly unique of course, but unique enough for my purposes). This value is given as the seed to a pseudorandom number generator. The first step, random(seed), shifts the x coordinate; the second, random(random(seed)), shifts y; the third rotates the curve handles, the fourth scales them (thus curve points always remain smooth and the handles keep their proportion). So, the randomization of a star's tip is guaranteed to remain the same so long as the tip itself remains unmoved; only as you start to move it, it starts to "dance" erratically (try it, it's fun). However if you move or transform the entire star with selector, the tip coordinates used in the above calculations do not change, and therefore the star is not re-randomized. It is also not re-randomized when you change the roundedness value, because that does not affect the position of the tip. Finally, the control handles show the positions of the true points, not randomized.
Overall, I think it's pretty convenient and useful. If you don't like a particular shape, just move one of the handles a tiny bit until you get something you like.
Actually, the most difficult part was to think up the solution to the problem that you're asking about. Once I invented the above method, it was a matter of an hour to code it. Worked fine at the first run.