lmer vs glmmPQL

?* Are you working in a different regime from previous studies
(smaller data sets, or some other point)?

I don't think so. The smallest setting has 10 groups with 5 obs. each,
the largest 500 groups with 20 obs. each, and the results indicate
that this is about the range in which interesting changes in mse/bias
happen for this (very simple) setting.

?* Does considering RMSE rather than bias give a qualitatively
different conclusion (i.e., PQL is biased but has lower variance)?

Yes, that's what I tried to display in the graphics -  for the
estimates of the random intercept sd and for data with m=10 or m=50
groups, PQL and LA/AGQ have about the same amount of bias but rmse is
(much) smaller for PQL.