What prints can I do to test/calibrate my printer for precision?
Any print you make can be considered a test print! But, specific test prints are easily found on the internet and lend themselves better for specific tasks. In order to improve your printer / print quality, you need to follow a meticulous series of tests and record the parameters you use to slice the model. Each test should be broken down into several print process parameters (and recorded) and tested for their effects (e.g. speed and temperature).

A general accepted strategy is to calibrate the extrusion process first and that look into the general accuracy.
Filament deposition calibration
An important aspect of printing is deposition of just enough filament (if you deposit too much the lines become wider, and so are the dimensions). To calibrate the extruder deposition of filament you can easily heat up the hot end, mark the filament with respect to a certain reference point and extrude e.g. 100 mm. Now measure the filament mark again with respect to the reference point to verify you actually extruded that same demanded amount.
Accuracy
For general accuracy printing of 20x20x20 mm cubes is usually fine. Note that the print accuracy is generally determined by the printer (construction) itself, but can be influenced by print speed. Large speeds in combination with a large mass of the printer head can lead to positioning inaccuracies (like overshoot). That said, you should also look into the printer itself. Does it have play? Are the belts tight enough, but not over-tightened? Is the bed skew, or are XYZ perpendicular?
This cube is frequently used to calibrate the dimensions:

Please do note that the poster of this picture probably produced a dimensional accurate print, but still faces under extrusion (as seen from the gaps between the lines)
If dimensions seem to be off reasonably, e.g. more than 0.2 mm (depending on the printer construction) you could change the amount of steps per mm in the firmware. Do note that the steps per mm is mechanically determined by the belt and pulleys.
More tests?
Even more tests can done to make specific filament adjustments. E.g. &quot;temperature towers&quot; can be printed to test the optimal parameters for filament flow and / or bridging (printing in air between two parts), or overhang (unsupported parts of the print). But, these last tests are not specifically designed for the issues you mentioned.
Tolerance test
Once you have calibrated the printer to the ability of yourself and to the capacity of the printer itself you could try to print a dimensional printing tolerance test. Such tests are usually constructed by cylinder shaped (or diabolic shaped) objects in a housing that is slightly larger that the shape.