Stability of Runge Kutta methods For the stability of RK methods, $y' = ky$ test function is used. What does this stability imply for general $f(t, y(t))$? What is useful meaning of stability and stability region?

Most constructs, like machines of robots, while having many moving parts, will have a very restricted overall motion repertoire. Meaning that apart from small deviations due to the inherent flexibility of parts, the whole construct will move inside a low-dimensional sub-manifold of the full state space.
Translated into ODE this means that many interesting systems, while having a high-dimensional state space, will rapidly converge toward some equilibrium  moving along a low-dimensional manifold. 
A numerical method should preserve this qualitative behavior. The common definitions of stability codify this for linear systems, with the expectations that non-linear systems can be locally linearized and that the error of the linearization is small enough over at least one step so that the linear stability also transfers to the non-linear system. When this fails, the combination of method and ODE is called "stiff". (Usually, a problem is stiff for all explicit methods if at all.)
