See attached, thanks!
Very nicely done Amanda! And nice substantive example.
One suggestion: usually even selection variables (eg S1, S2) are assumed to be determined with error, ie they are not a deterministic function of X and Y but rather a function of X, Y, and some random noise. This would generally weaken the magnitude of selection bias.
For calculation of type 1 error rate under the null, you need to save your estimated coefficients and CIs or p-vals from each bootstrap replication (this is an option in the bootstrap command) and see what fraction of the iterations have a p<.05. But yes, you are probably right that you'd always have a 100% type 1 error rate.
Finally, we usually bootstrap replicates w/ the same sample size as the original sample.