Vibration following an impact

Regarding your first question IMHO eigenvalue analysis is appropriate and valid as long as the material is within the elastic range. Also another misconception is that, the application of eigenvalue analysis depend on how close the excitation frequency to the fundamental frequency. (In actuality, IMHO you can regard the cymbals case as an impulse, not as a harmonic excitation).

Regarding the second part of the question:

My guess would be that the initial response to the impact is transient (wave propagation) and would eventually converge to the combination of modes predicted by the eigenvalue analysis. Is that correct? If so, what influences the speed of this convergence?

Eigenvalue analysis is not about the steady state response. In many aspects, it is mainly about the transient response. Somehow I got the feeling that you interpret, the eigenmodes (or mode shapes) like standing waves (which in my opinion they are not). IMHO, eigenvalue analysis is about expanding the concept of the natural frequency of a SDOF to a MDOF or continuous system.

In the SDOF system, the transient response is depended on the natural frequency, and the steady state response is depended on the excitation frequency. So, if you extend that to an MDOF, the natural frequencies are dominant on the transient response. Also, since the excitation of the cymbals is impulsive (not harmonic), their response also qualifies (IMHO) as transient.