I have some predictors in my model with missing values, so I sample the missing values from a parameter vector and create a new predictor matrix combining my observed data and sampled values. When I check the output later, the observed data elements of the matrix show very low (e.g, 2) neff. When I plot the values, they are a flat line, as expected. Is this just a summary issue, or does whatever numerical fun is occurring here have implications for sampling also?

You don’t need to worry about the diagnostics for fixed/known values. You can also move the contstruction of that matrix to the model block and avoid saving those known elements entirely, assuming that you just want the matrix for convenience in the model block, which is often the case in this situation.

Right – Rhat and ESS, or at least their estimators, are ill-defined for constants.