Condition for modelling noise as white noise
I will try to answer your three questions as a whole, with the following insights:

"Whiteness" means: flat power spectral density, thus zero autocorrelation for lag different than zero (uncorrelated samples), thus wide sense stationary (time-independent autocorrelation and mean).

White Gaussian noise passing through a linear filter always produces Gaussian noise, only changing its parameters (mean and variance), since any linear combination of uncorrelated Gaussian random variables produces a Gaussian random variable. However, the whiteness is lost in general, and thus samples become correlated after passing a linear filter.

If your system's transfer function has a bandwidth in $\omega$ significantly lower than the noise's bandwidth, you may approximate the input noise as white, since what the system "sees" is a flat spectrum.

Anyway, you must take into account that white noise is always an approximation for real-life systems. There is no such thing as a noise with infinite bandwidth in practice. But you may be in presence of noise with very large bandwidth (compared to your system), and thus successive samples of such a noise are (nearly) uncorrelated.