When you say "white" you assume it's WSS to begin with. For non-WSS processes, "white" isn't defined, since no only lag-dependent autocorrelation can be found. (And a process is white, exactly if its autocorrelation takes the form of a delta dirac impulse.)

So, yes, any process that is called "white" is inherently WSS.

"Gaussian white noise" is white noise whose amplitude is Gaussian-distributed. Amplitude distribution has nothing to do with whiteness or stationarity: a non-stationary process can still be Gaussian distributed at any point in time.

Disclaimer: this might very well be wrong. Still pondering it, but Dilip Sarwate has convincing points.

When you say "white" you assume it's WSS to begin with. For non-WSS processes, "white" isn't defined, since no only lag-dependent autocorrelation can be found. (And a process is white, exactly if its autocorrelation takes the form of a delta dirac impulse.)

So, yes, any process that is called "white" is inherently WSS.

"Gaussian white noise" is white noise whose amplitude is Gaussian-distributed. Amplitude distribution has nothing to do with whiteness or stationarity: a non-stationary process can still be Gaussian distributed at any point in time.