Can spectral density be a complex quantity?
and the same goes with the spectral density (Fourier transform of the auto-correlation function). 


No, that's not the case.

Since the autocorrelation is a hermitian symmetric function for any $S$, its Fourier transform is always real.


  If the above analysis valid? as the process is not wide sense stationary. 


If the process is not WSS, then you can't just proclaim $E[S^*(t)S(t-\tau)]$ to be dependent on only one variable (usually, $\tau$), and hence, a (1D) Fourier transform doesn't make much sense.


  If the analysis is not valid, is there some way to handle this kind of situation?


Depends! You might want to define/find coherency times and do Short-Time Fourier Transforms within that.

Your system, in fact, is just a phase shifted impulse response – as such, a phase-delay representation, which might be derived from a Frequency Shift-Delay plane, might be more helpful in analyzing things. You'll find such a Frequency Shift-Delay plane in what is called scatter function in wireless communications, representing the Doppler and path coefficients of a wireless channels in motion.

But in your case: Is trying to understand PSD or PSD-equivalents really useful? Don't you just want to build a parametric estimator for $\mu_G$, $\sigma_G^2$ and $\theta$ instead?