Thanks for your interesting questions about the edgeR model.
1) We assume that variability in RNA-seq counts come from three sources:
a) sampling variability associated with sequencing (for each lane),
b) technical variation in library preparation (lane to lane), and
c) biological variation
Sources (b) and (c) affect the underlying concentration of each transcript in each RNA sample, whereas (a) affects the precision with which this concentration is measured by sequencing technology. The dispersion parameter in the edgeR model measures the squared coefficient of variation (CV) of each transcript's concentration arising from sources (b) and (c). Experiments suggest that variability from source (b) is relatively minor, so the dispersion is essentially the squared CV of biological variation.
If you sequence deeply enough, you can theoretically eliminate variability from source (a) almost entirely. In other words, you can determine almost perfectly the concentration of each transcript in each sample. However you can't eliminate biological variability (b+c) in this way. As you sequence more and more deeply, power to detect differential expression is eventually determined only by biological variation, hence the asymptote that you mention. In the edgeR model, this is reflected by the fact that observed transcript concentrations converge to gamma distributed random variables with CV = sqrt(dispersion).
To further increase the power to detect differential expression you would need to reduce biological variability as well, and you could only do that by increasing the number of biological replicates. This is what the model predicts.
2) When shrinking the dispersion estimates, the amount of shrinkage depends on the precision with which the original value is estimated as well as by the weight of the prior distribution. For a given number of libraries, larger counts give more reliable estimates of the dispersion than small counts. Hence dispersions for rare transcripts tend to be shrunk more than dispersions for very abundant transcripts. Hence the shrinkage is not monontonic.