Is there any well-known test set of 3D models that I can print out which will clearly show up the problems I need to work out in calibrating my 3D printer? Ideally, it would be either one or multiple models which shows the point at which overhanging starts to fail, points at which stringing occurs, the accuracy of one layer over another, and maybe edges which are supposed to be a particular length.
There are more test models one can download than would be possible to list here. A search on Thingiverse results in pages and pages of useful models. A common model for testing is called the 3d Benchy, although it is not particularly parametric. Overhangs, retraction, layer alignment and other aspects of your printer are tested with this model.
Another option is to purchase a set of test models from 3dkitbash.com:
The models lists as such:
#3dk01 - Voids #3dk02 - Spikes #3dk03 - Flex #3dk04 - Edge #3dk05 - Text #3dk06 - Bridge #3dk07 - Dissolve #3dk08 - Ball Joint #3dk09 - Hinge #3dk10 - Columns
which are more specifically defined for testing your printer. I would expect that Thingiverse and other model-sharing sites would have similar designs available to meet your specific requirements.