Understanding the bike geometry through riding
What would be some good examples of test cases to assess the impact of
  these parameters?


Build a testbed bike that has adjustable parts covering the different parameters you care about. Instrument it appropriately and measure the changes to the factors you care about as you vary the parameters. This is necessarily vague because there are upwards of 10 variables that affect more than 10 output parameters. This is the reason your question is being marked as "too broad".

For example, a number of people have built bike frames with variable headset angle and attached front forks that have variable axle offset. That allows much experimentation with rake and trail. But the outputs measured are typically "how it feels", because it's very difficult to measure the second and third order effects you're likely to care about, especially since the primary measurements (typically 6 axis acceleration) are very noisy.

Another reason your question is too broad is that each of the questions implied above has been the topic of multiple PhD research projects, from "how do I smooth the output of my accelerometer" to "how much slop is acceptable in the bearings of my headset".


  My aim is to be able to ride a bike, and exactly pin point to the changes 
  that need to be made in a particular order to meet certain expectations.


If you can't precisely specify the expectations it will be impossible to meet them. You need numbers here, and both precision and accuracy. Ending up with "head angle must be 78.24°" means you need a minimum of four significant figures at every stage right from the expectations. So "I require not more than 4.123N of input force at the centre of the handlebar grip to maintain the centre of mass in a straight line path when hitting a 45.01° barrier 100.4mm high at 12.62m/s" would be an example of one test case or "expectation".

There's a book called "Lords of the Chainring" that analyses all these factors in combination, and it is arguably the definitive work on the topic (I don't think it's as complete as the author does, but there is nothing better that I'm aware of). The author used to teach an engineering course in the subject and has extensive experience both theoretical and experimental (from years of students experimenting and reporting results). 

I have a copy and have used it to design bikes, but I decided that in most cases it was more effort than could be justified for my garage builds. I still use it to get an idea of the desirable values for a particular design, but I've also discovered that I can comfortably ride bikes that are on the fringes of what Bill says is stable. This might not be true for you.