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Are the pole-to-equator temperature gradients lower at higher heights than at lower heights (like 850 mb/500 mb)?
If so, why is it the case? Especially given that the zonal circulation tends to be much higher at higher heights, so one could expect there to be less meridional heat transport.
(Bonus question: how is it different in other planetary atmospheres?)
It depends on the season. The figures below shows zonal mean temperature for June-July-August from ECMWF ERA-40 reanalysis. As you can see, at 100 hPa, the equator is actually colder than the sunlit hemisphere.
For getting the "facts" on this kind of questions, the ECMWF ERA-40 atlas is a good source. Now as to why this is the case; that is a much more involved question on atmospheric dynamics and circulation. The Earth heats the surface, in particular where it's sunlit. It also heats the spring and summer stratosphere, where ozone absorbs solar radiation. Heating causes air to rise from the surface to the tropopause, where it's forced poleward creating a Hadley cell. Meanwhile, the Earth rotates and everything becomes a highly chaotic and dynamical system. So the fact that winds are strongest near the tropopause (see, for example, DJF winds) is not a direct consequence of where the heating occurs, but rather the effect of a lot of dynamics.
For further reading on why atmospheric circulation is as it is, one needs to study textbooks. Some famous textbooks on dynamic meteorology and atmospheric science are:
