How does the pole-to-equator temperature gradient scale with height contours?
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.

June-July-August



December-January-February



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:


James Holton, An Introduction to Dynamic Meteorology
Wallace &amp; Hobbs, Atmospheric Science: An Introductory Survey