My simple head space (as I was taught and re-learned thru experience, and have passed on)
1. Kalman Gain close to 1 or 0 is a warning sign that careful consideration is needed.
This fact can be brought up immediately in example #5 and continued
2a. K close to 1.0 can be bad because..., however for some applications (dynamic models) it can be acceptable since...
2b. K close to 0.0 can be bad because... however for some applications (dynamic models) it can be acceptable since...
3. To solve the problem from step 2, As a first step, for those applications where K close to zero or one is bad... a fudge factor term (called Q for reasons discussed later) can be added to the Kalman Gain computation
3a. Choosing the correct fudge factor for the application is often very difficult and may require lots of simulation runs (a parameter study) with different measurement sequences (including some expected off-nominals) and various values for the process noise.
Remember we are designing a filter, likely for a new application (or a non-trivial extension of an existing application)... so all the elements of an engineering design are needed. Make solution hypothesis, test them, refine them, test them some more with greater realism and eventually real-world data, continue to refine the solution.
4. For easy case of a simple application and only a few unknown states, the process noise can be guesstimated from experience. For more complex applications (perhaps there are dozens of unknown states to estimate) a more rigorous approach to select the correct mathematical description of Process Noise is needed.
-- End of Fudge Factor discussion --
{I think you covered this section well} Then you can introduce the notion that the state dynamics cannot model everything and that unmodeled part can be approximated by Process Noise. For example an unmodeled constant acceleration, gives a process noise of ....
Here are some sentences I think are wrong or misleading
"As you can see, the Kalman Gain gradually decreases; therefore, the KF converges." However, the Kalman Filter may converge to garbage. This garbage could be a "lag", or just plain wrong.
"The process noise produces estimation errors." A well chosen process noise is important to reduce estimation errors over an ensemble of conditions, by accommodating a range of unmodeled state dynamics. A poorly chosen process may not improve anything.