Gosh, thanks! I check this answer like every 2 days :)

Definitely not a synonym for heatmap! Heatmaps are typically about creating a raster interpolation of the density of observations given point data. Hotspot analysis is about locating areas of statistically significant geospatial clusters of observations with a particular attribute within a larger sample population. Put more simply, if we were mapping crime, a heatmap could let us see where crime is taking place, full stop; a hotspot analysis could show where crime is taking place at higher than expected levels given the existing population.

just echoing @ndawson for anyone else stuck here. I was running 2.4 and had this problem - upgrading to 2.14 solved it immediately

Thanks, that was exactly it. Similarly, seeing now that raster.meta shows (among other things): 'nodata': -3.4e+38

just in case anyone else gets stuck, Zonal Statistics is a plugin that you have to install/enable via the plugins menu - otherwise it won't be present in the raster menu...

oh my goodness this is maybe the most pain an answer on stackexchange has ever spared me