Hi Bob!
I can’t answer your first question, but it’s interesting, I’m curious about the answer!
As for your second question, I’m not sure, but I don’t think you can do that: the point of the non-centered parametrization is to get the average effect (mu) out of the prior. tau * theta_tilde must then just be a deviation from the baseline.
But if tau * theta_tilde also contains a baseline value, then the model will be overparametrized in my opinion – an infinite number of different values of mu and tau * theta_tilde will produce the same sum mu + tau * theta_tilde.