Why is White Noise so important in System Identification or Adaptive Filters
I believe the point of feeding white noise into the system is for the filter to adapt its coefficients before actually generating the signal $x[k]$. This would mean there are two "operating modes" for the system: coefficient adapting mode (in which white noise, a broadband signal, is used to adapt the filter to the feedback path), and performing mode (where the signal $x[k]$ is input into the system with the filter already adapted).

So the answer to your first question would be that the noise is output by the loudspeaker and follows the whole signal path. The system adapts it's coefficients to be similar to $F(q)$, and once that process has ended the white noise can be turned off. Afterwards, when generating $x[k]$ the system will already reduce feedback using the $\hat{F}_0(q)$ transfer function that was adapted previously, and the coefficients will only change when the forward path $G(q)$ is modified.

The purpose of using white noise (or a pulse train) is to have a broadband signal that has frequency components throughout the whole audible spectrum so that the adapted filter works for any kind of input signal $x[k]$. If the system was used without first feeding the broadband signal into it, every time $x[k]$ presents new frequency components, the coefficients may need to adapt during the performance/speech, which may have audible effects depending on the convergence speed of the algorithm.