Understanding this spacetime diagram

That is a conformal diagram representing the Haggard-Rovelli spacetime. The thick line represents the collapsing/bouncing null shell. The region I is enclosed in the shell and for the Birkhoff theorem, it is Minkowski. The region II is a portion of the Kruskal extension of Schwarzschild. The peculiar thing about conformal diagrams is that the causal structure is preserved or in other words that null trajectories are always at $45^{o}$ degrees. The region III (and the corresponding symmetrical one above $t = 0$ ) is the one where the Einstein equations are no longer valid. The metric, due to spherical symmetry, has the same form in all the region $$ds^{2} = -F(u,v)dudv + r^{2}(u,v)(dtheta ^{2} + sin^{2}theta dphi^{2}) equiv bar{g}_{mu nu}dx^{mu}dx^{nu}$$ where $F(u,v)$ and $r(u,v)$ needs to be glued at the boundary of the regions (like in the eq.s (27)-(30) of the paper you linked). The collapsing is spherically symmetric, so in general, there are not gravitational waves. If you instead perturb the quantum region you can draw gravitational waves as photon shells and study the GW propagation in the Regge-Wheeler approach $$g_{mu nu} = bar{g}_{munu} + h_{munu}$$ I think your question is more general and concerns the propagation of GW in general backgrounds. You can find a reference in these two papers: https://arxiv.org/abs/1503.08101 and https://doi.org/10.1103/PhysRev.108.1063.