Reply to "side shaft play"

There's really no place where side-to-side play can develop except in the tapered bearings on the ends of the differential assembly, possibly from lack of preload. And I can't think of anything that would reduce preload except maybe using the wrong shims in the side plates, or a too-thick non-stock side cover gasket. Did someone work on this ZF to stop some leaks, perhaps?
If not, a popping noise at very low speeds can sometimes result from using 70W90 diff lube WITHOUT posi-lube. The limited-slip clutch plates stick together and when they pop loose, you might hear a noise. I always recommend using diff-lube WITH posi-additive in it.
Second possibility: if even one of the 16 halfshaft bolts are slightly loose, it can and has caused a noticable vibration under severe acceleration. I suppose its possible one might hear this looseness if pushing it around by hand. I first discovered this on a friend's car; close exam showed a cracked lock-washer in which most of it had fallen out leaving a gap between the companion-flange and the bolt head. A few weeks later, another owner that was present found one bolt on his Pantera that was loose. Both car's vibrations were completely fixed by simply tightening up all 16 bolts.