About the variance of the coefficient of narrowband noise when using signal-space representation
Since the basis functions are narrowband, the formula still works and the variance of each $n_j$ remains $N_0/2$..

In other words, assume the bandwidth of the basis functions is $B$. You can then think of them as filters that ignore any frequency outside of $B$. So, they "don't care" if there is noise outside of $B$ or not.

The expression for $n_0(t)$ will change, though.