How should I use rotary switches and resistor networks to uniquely relate each possible combination of switch settings to an analog voltage?
If you wire all three switches to give a single analog output, you require 0.1% resolution.  Since common precision resistors are 1%, and ADCs typically have a couple of counts of uncertaintly, I don't think this technique is practical.

However, if you wire each switch independently to an ADC input, with resistors selected to give evenly spaced voltages, it would be much easier to get reliable readings, and would even work with 5% resistors.

You could use three ADC inputs, or a single ADC with analog switches to read the input switches one at a time.