I like to discuss too, although the more I study stuff like this the more I realize I don't know...
Tire spring rate affects mechanical spring rate in a "parallel" fashion, despite it looking like the two spring rates are connected in series.
When you connect any number of springs this way, and use "k" as each spring's spring rate, the reciprocal of of the final total (equivalent) spring rate equals the reciprocals of each individual spring rate all added together. As a mathematical formula:
1/k(eq) = 1/k1 + 1/k2 + 1/k3 ... (continued for as many series connected springs that you care to involve)
If you are only considering two spring rates (the tire and the mechanical spring) then you can do some algebra and obtain this formula:
k(eq) = (k1 x k2) / (k1 + k2)
However, if your car has bump stops then the first formula allows you to include them too. Note that because bump stops are thin they only effect the overall spring rate for a short amount of travel (until they're fully squished) and then for the rest of the system's travel distance they drop out of the equation.
If you know your chassis and suspension components can flex then those tend to add directly to the overall system spring rate (because they act physically in parallel to the tire / mechanical spring assembly).
Generally, the spring rate of the tire is less than the spring rate of the mechanical spring so the tire dominates the system. You can plug some numbers into the formulas to see their effects. And generally as tire pressure goes down so does the tire spring rate. A little tire pressure change can cause a big tire spring rate change - that's why some racers carefully adjust it to a tenth of a pound.
By the way, if your stock car is sprung so stiffly that it acts like a go-kart then flexing in the chassis components becomes important. In karts we include spring rates of the axle portions that each tire bolts to, as well as flexing of the frame. Racing kart chassis are purposely built with complex angles and differing tubing sizes and thicknesses to use the frame as a spring.
Now about the shock absorbers, I prefer to think of them as dampeners rather than like a hand dribbling a ball - because shocks react to the forces applied to them while that hand is imparting force into the dribbling system. But there's no foul in considering that a bouncing wheel could act like a hand... just need to also consider that a dribbling hand tends to be more consistent than a wheel bouncing on an irregular surface. I also prefer to think of the wheel / mechanical spring system as being contained by mounting hardware, so instead of not knowing where it will go if uncontrolled you do know the plane of movement for the bulk of the system's energy. However, I concede that some energy does move laterally (springs can bow side-to-side and a tire sidewall definitely bows all over the place) so there is no harm with your control analogy. Overall though the bulk of a shock's control forces are in the direction of the shock shaft sliding in and out of the shock body.
