1. On the maximally extended Schwawrzschild solution there is no decay to the wave equations: the space-time "ends" in finite proper time at the singularity, and there's not enough time (compare to the not-enough-space scenario you mentioned with compact Cauchy surface) for solutions to decay.

2. Compact Cauchy surface does not rule out decay. Wave equation on de Sitter space is known to decay (strictly speaking, not in $L^\infty$ due to the constant solutions; but in $\mathring{W}^{1,\infty}$).

3. The results concerning Schwarzschild and Kerr mostly deal with their outer domain of dependence (with some authors considering a little sliver that pokes into the black hole). If you take a steep time function on the outer domain of Schwarzschild, for example (and an example would be the function $t$ in the traditional Schwarzschild coordinates), the solutions to wave and Klein-Gordon equations have absolutely no decay even for compactly supported initial data, as a wave packet can fall into the black hole very quickly, but the black hole boundary is "at $t = \infty$".

   If you actually read the papers on Schwarzschild and Kerr decays, you will see that instead of using a Cauchy time function, their "time function" when measuring decay are never Cauchy. It usually intersects the black hole boundary (and sometimes also $\mathscr{I}^+$) transversely.

4. In fact, the only cases where I can think of where decay of wave/Klein-Gordon equation has been proven for a globally hyperbolic Lorentzian manifold with respect to a steep time function are (1) background metrics which are perturbations of Minkowski space [where generic perturbations cannot improve things since Minkowski is as about as good as it comes]; (2) cosmological scenarios [independently of whether you have an open or closed Cauchy hypersurface] where the energy decay is driven by the spatial expansion and so is independent of trapping or caustics (which capture failure of "dispersion to infinity"); and (3) works on static product metrics where one can do resolvent estimates.

I admit not being up to date in the results of type (3), but it seems that the type of hypotheses usually assumed for obtaining results of type (3) are generally stable under perturbations.

5. The focus on effects of trapping and caustics that you mentioned in your question is mainly due to them being obstacles for (global and local) dispersion. So your question really only makes sense when the decay effects are dispersion dominated (see my comment in the previous point about cosmological solutions). As such you probably want your manifold to admit global dispersion in a suitable sense, which you can guarantee by, e.g., imposing some sort of asymptotic simplicity or asymptotic flatness.

6. On the other hand, for wave equations in particular, it has been long understood that the decay parametrized by a time function $\tau$ does not capture the real underlying phenomenon where the decay should be understood more along the line of something like "parametrized by affine parameters of a null geodesic congruence",[insert place holder here; on second reading I am not 100% sure about my word choices there; until I can come up with a better way to say it succinctly, I'll leave this at "something technical to describe"] which incidentally also more closely resembles the result from the stationary phase argument if you look at the geometric optics. This is related to why in point 3 above, the "time function" used to measure decay in modern literature are not Cauchy time functions.

7. I am also not sure what you mean by that trapping should disappear under generic perturbations. Consider the class of (connected) globally hyperbolic Lorentzian manifolds with two asymptotically flat ends. This class is stable under small perturbations. But such manifolds must have trapped null geodesics.

1. On the maximally extended Schwawrzschild solution there is no decay to the wave equations: the space-time "ends" in finite proper time at the singularity, and there's not enough time (compare to the not-enough-space scenario you mentioned with compact Cauchy surface) for solutions to decay.

2. Compact Cauchy surface does not rule out decay. Wave equation on de Sitter space is known to decay (strictly speaking, not in $L^\infty$ due to the constant solutions; but in $\mathring{W}^{1,\infty}$).

3. The results concerning Schwarzschild and Kerr mostly deal with their outer domain of dependence (with some authors considering a little sliver that pokes into the black hole). If you take a steep time function on the outer domain of Schwarzschild, for example (and an example would be the function $t$ in the traditional Schwarzschild coordinates), the solutions to wave and Klein-Gordon equations have absolutely no decay even for compactly supported initial data, as a wave packet can fall into the black hole very quickly, but the black hole boundary is "at $t = \infty$".

   If you actually read the papers on Schwarzschild and Kerr decays, you will see that instead of using a Cauchy time function, their "time function" when measuring decay are never Cauchy. It usually intersects the black hole boundary (and sometimes also $\mathscr{I}^+$) transversely.

4. In fact, the only cases where I can think of where decay of wave/Klein-Gordon equation has been proven for a globally hyperbolic Lorentzian manifold with respect to a steep time function are (1) background metrics which are perturbations of Minkowski space [where generic perturbations cannot improve things since Minkowski is as about as good as it comes]; (2) cosmological scenarios [independently of whether you have an open or closed Cauchy hypersurface] where the energy decay is driven by the spatial expansion and so is independent of trapping or caustics (which capture failure of "dispersion to infinity"); and (3) works on static product metrics where one can do resolvent estimates.

   I admit not being up to date in the results of type (3), but it seems that the type of hypotheses usually assumed for obtaining results of type (3) are generally stable under perturbations.

5. The focus on effects of trapping and caustics that you mentioned in your question is mainly due to them being obstacles for (global and local) dispersion. So your question really only makes sense when the decay effects are dispersion dominated (see my comment in the previous point about cosmological solutions). As such you probably want your manifold to admit global dispersion in a suitable sense, which you can guarantee by, e.g., imposing some sort of asymptotic simplicity or asymptotic flatness.

6. On the other hand, for wave equations in particular, it has been long understood that the decay parametrized by a time function $\tau$ does not capture the real underlying phenomenon where the decay should be understood more along the line of something like "parametrized by affine parameters of a null geodesic congruence",[insert place holder here; on second reading I am not 100% sure about my word choices there; until I can come up with a better way to say it succinctly, I'll leave this at "something technical to describe"] which incidentally also more closely resembles the result from the stationary phase argument if you look at the geometric optics. This is related to why in point 3 above, the "time function" used to measure decay in modern literature are not Cauchy time functions.

7. I am also not sure what you mean by that trapping should disappear under generic perturbations. Consider the class of (connected) globally hyperbolic Lorentzian manifolds with two asymptotically flat ends. This class is stable under small perturbations. But such manifolds must have trapped null geodesics.

5. The focus on effects of trapping and caustics that you mentioned in your question is mainly due to them being obstacles for (global and local) dispersion. So your question really only makes sense when the decay effects are dispersion dominated (see my comment in the previous point about cosmological solutions). As such you probably want your manifold to admit global dispersion in a suitable sense, which you can guarantee by, e.g., imposing some sort of asymptotic simplicity or asymptotic flatness.

6. On the other hand, for wave equations in particular, it has been long understood that the decay parametrized by a time function $\tau$ does not capture the real underlying phenomenon where the decay should be understood more along the line of something like "parametrized by affine parameters of a null geodesic congruence", which incidentally also more closely resembles the result from the stationary phase argument if you look at the geometric optics. This is related to why in point 3 above, the "time function" used to measure decay in modern literature are not Cauchy time functions.

7. I am also not sure what you mean by that trapping should disappear under generic perturbations. Consider the class of (connected) globally hyperbolic Lorentzian manifolds with two asymptotically flat ends. This class is stable under small perturbations. But such manifolds must have trapped null geodesics.

5. The focus on effects of trapping and caustics that you mentioned in your question is mainly due to them being obstacles for (global and local) dispersion. So your question really only makes sense when the decay effects are dispersion dominated (see my comment in the previous point about cosmological solutions). As such you probably want your manifold to admit global dispersion in a suitable sense, which you can guarantee by, e.g., imposing some sort of asymptotic simplicity or asymptotic flatness.

6. On the other hand, for wave equations in particular, it has been long understood that the decay parametrized by a time function $\tau$ does not capture the real underlying phenomenon where the decay should be understood more along the line of something like "parametrized by affine parameters of a null geodesic congruence",[insert place holder here; on second reading I am not 100% sure about my word choices there; until I can come up with a better way to say it succinctly, I'll leave this at "something technical to describe"] which incidentally also more closely resembles the result from the stationary phase argument if you look at the geometric optics. This is related to why in point 3 above, the "time function" used to measure decay in modern literature are not Cauchy time functions.

7. I am also not sure what you mean by that trapping should disappear under generic perturbations. Consider the class of (connected) globally hyperbolic Lorentzian manifolds with two asymptotically flat ends. This class is stable under small perturbations. But such manifolds must have trapped null geodesics.