Cross correlation and cross power spectrum for signal + noise
$S_{xy}=S_{xx}$ is true but this does not mean $F_x*F_y = F_x*F_x$ when $y$ is random. 

The actual definition is $$S_{xy}(f)=\lim_{T\rightarrow \infty}\textbf{E}\{F^*_xF_y\}$$ for random signals. Since $x$ and $v$ are uncorrelated, this would be equal to $$\lim_{T\rightarrow \infty}\textbf{E}\{F^*_xF_x\} = S_{xx}.$$ But your next assumption is not correct.

$F_x$, $F_y$ are only based on single realization of $x$ and $y$ respectively. For random signal, $S_{xy} \ne F_x^*F_y$