Like Happyman, I was under the impression that they were "jumping" miles high into the air. Now assuming they used the same lashings as Szeth, it is possible that they reduced their weight just enough so that a Plate powered jump would have this effect. I, however, saw it as more of a blast, shifting gravity completely around for instance.
So the plate probably didn't store stormlight then. However, Dalinar didn't mention any inhaling of the stuff. Maybe the plate just helps them retain it and not be so leaky. I mean, If it didn't constantly dissipate, you could go into battle completely full. Perhaps the Plate itself was an effort to mimic the voidbringers who held in stormlight perfectly.
As for how the Spheres are made, it would be simple in a world where soulcasting exists. Just wrap a gem with clay and soulcast to glass, or whatever they are made of.