Great post, quite a lot of stuff here.

One of the big questions about Pi is the normality of its digit sequence (essentially how random are the digits) As far as I know, the digit sequence to all irrationals including Sqrt[2] are essentially random (see the discussion in chapter 4 of Stephen Wolfram's A New Kind of Science )

Do Michael's results on the dates show any sign of regularity? If we replace the pi digits by a random sequenceall of the pictures should look similar.

Is there a Random Day?

Another thing to wonder about is the distribution of birth days, which is complicated because it is not uniform for the general population, but the question is whether birthdate correlates with the likelihood of taking part in PiDay.