Why do you expect the normalized operator, as you have defined, to be nonzero? Shouldn't it have a zero at the special representations, i.e. $\chi_1\chi_2^{-1}=p^{\pm1}$? Most probably to get a non-vanishing holomorphic operator one normalizes by $\epsilon(0,\chi_1\chi_2^{-1})\frac{L(1,\chi_2\chi_1^{-1})}{L(0,\chi_1\chi_2^{-1})}.$

Can one say that the Plancherel constant (density) defined by Shahidi is actually analytic continuation of the Plancherel density for the tempered representation?

Yes this is known to me. But why should that formula, which is for the symmetric spaces, extend to the group case?

I might be wrong but I am confused with your "impression". Take $G=SL_2(\mathbb{R})$. The discrete series representations basically consist of various (non-zero) weight modular forms, where as the principal series representations contain Maass forms which are part of the discrete spectrum of $L^2(\Gamma\backslash G)$. So "discrete series representation" is somewhat smaller than "discrete spectrum in $L^2$".

Dear GH, is there any Dirichlet type approximation in the format of Frustenberg, i.e. $|2^m3^n x-p|<\dots(o(1))$ for some integer $p$?

Could you please tell me the exact statement of Berard's theorem of $\log$ saving?

Thank you very much for such an illuminating discussion!