Hopefully some of this answer will be helpful in attempting to answer the "when am I allowed to perform [a Wick rotation]," part of your question. As to what carries over, well after my description, it should seem that all observables are, but most definitely not the group representations (nor positivity).
With the viewpoint of the path integral, when we are computing an $n$-point correlation function $G(t_1,...,t_n)\equiv G_n$ (where I am only labeling the time-component for the sake of this answer) from the path integral, we need to ensure the convergence of it. To do so, it is useful to analytically continue all the time coordinates that we order by hand $t_1<t_2<\cdots t_n$, to complex time with the specific ordering of imaginary time as \begin{equation} \text{Im}\:t_1<\cdots <\text{Im}\:t_n \end{equation} so that analyticity is maintained. The real parts of $t_n$'s are not subject to any ordering a priori.
It is convienent to start with the Euclidean correlator \begin{equation} G_n^E\equiv \langle\Omega|\hat{\phi}(t_1)\cdots\hat{\phi}(t_n)|\Omega\rangle\equiv \langle \hat{\phi}_E(t_1^E)\cdots \hat{\phi}_E(t_n^E)\rangle \end{equation} where $t_i = -i(t^E_i)^D$, $\mathbf{x}_i\equiv \mathbf{x}_i^E$ with ordering $(t^E_1)^D>(t_2^E)^D>\cdots >(t_n^E)^D$. What this does is that every imaginary time component is rotated to the real axis or becomes the Lorentzian time via \begin{equation} -it_i = e^{-i\alpha}t_i^L \end{equation} where $t^L$ is the typical Lorentizian time (I should have used $\tau$ and $t$...). Notice that if we set every rotation angle $\alpha = \pi/2$, then this maintains the ordering of $\text{Im}\:t_i$. This uniform Wick rotation for all $i = 1,...,n$ leads to the Lorentzian correlator. And in the limit $\alpha\rightarrow 0^+$, we have a real-time correlator that is time-ordered.
The conclusion is then that the time-ordered real-time correlator is obtained from the Euclidean one by a uniform Wick rotation on all $t_i$'s.
So I would argue that a Wick rotation can always be performed when the ordering of the Euclidean time components can always be ensured. However, even then there are subtleties to this since if you perform the Wick rotation from the Euclidean 2-point Green's function for scalars and keep a general angle $\alpha$, you get a denominator as $(-e^{2i\alpha}(k^0)^2 + \mathbf{k}^2+m^2)^{-1}$. If you set $\alpha = 0$, the integrand in the denominator will vanish when $k^0$ hits $\pm\sqrt{\mathbf{k}^2+m^2}$. To maintain analyticity on the Lorentzian side, this is (one) reason we add $-i\epsilon$ to the denominator.
Out of this we have 3-correlators: the Euclidean correlator which when analytically continued is technically the Wightman function, the Wightman function as mentioned, and then from the Wightman functions we can build the time-ordered Green's function.
And so the invariant quantities when Wick rotating are essentially all observables, but not the representations, and hence particles should only be defined in a Lorentzian manner. And thus I would suspect that only quantization in the Lorentzian manner is proper, not in Euclidean space.
Also, this is also when thinking about Minkowski spacetimes, and there are extreme subtleties when dealing with curved backgrounds. It is also in this viewpoint that I really only see Wick rotation as a helpful computational tool, and that's it.
