In the framework of the Kalman filter
1) How can we check errors are Gaussian and independent?
Check out this answer for a way to check for independence (whiteness).
This answer gives one way (not necessarily a good way) to check for Gaussianity: calculate the kurtosis and check that it's close to 3.
2) If the processes are not stationary, could that be a problem?
Only if your signal model (state transition matrix, input matrix, and output matrix, process noise covariance, measurement noise covariance) do not reflect the sort of nonstationarity that your signal has.
3) If I have the unobservable variables that are latent. How could I get the transition matrix or the covariance matrix for the equations?
How do you get the transition matrix ever? Make it up!
How do you know that there are unobservable states if you don't already have a signal model?
