Properties of Stochastic Processes
As pointed out in the comments, this question is rather broad, so here I will just answer your last question about the autocorrelation of the output signal $y[n]$. You might know the following relation concerning the power spectra of input and output sequences:

$$S_y(\omega)=S_x(\omega)|H(\omega)|^2\tag{1}$$

where $H(\omega)$ is the frequency response of the LTI system. From the inverse Fourier transform of (1) you get a corresponding relation between the autocorrelation functions:

$$\phi_y[m]=\phi_x[m]\star h[m]\star h^*[-m]\tag{2}$$

where $h[m]$ is the system's impulse response, and $\star$ denotes convolution.