How does sampling rate impact Discrete-Time Kalman Filter state space modeling assumptions?
The answer to your question is : maybe. When one is at the discrete time model, it is generally assumed that you solved the state transition equations from the continuous time differential equations where the physics have already been appropriately modeled, which is where your answer lies. Time discretization is based on sampling the continuous time state transition matrix, thus the sampling intervals aren't related to Nyquist, they correspond to the fidelity of the modeling. Shorter intervals tend to be more accurate approximations. The time intervals don't have to be uniform, but uniform intervals provide a less complicated implementation. 

While shorter time intervals correspond to a heuristic where shorter has better approximation, you raise a valid concern, sampling too closely, may indeed introduce correlation in the process noise.  This can be solved with an augmented state. 

These are all considerations. Kalman Filters need to be tuned.  The measurement noise has been neglected in this discussion.

So, the answer to your question is that it really depends.