Ok, I'll do my best to help out here.
1) StuG IV's first appeared in December '43 and served on all fronts from that point forward. So you can consider Italy, the Eastern Front, and the West after Normandy as viable time periods.
2) StuGs were assigned to many units but usually in their own battalions...and there were many battalions formed throughout the course of the war, far too many to list here. The StuG IV didn't receive any special consideration when assigned to a unit...it was after all designed as companion to the StuG III when StuG III production was interrupted by Allied bombing in 1943.
3) Stug IVs followed the same design timeline as StuG IIIs, so as StuG III features were introduced, they also appeared on StuG IVs. Zimmerit would've been present on StuG IVs produced from Dec '43 through Aug '44, so highly plausible to find one with zim also sporting whitewash in either the 43-44 or 44-45 winter setting. Schurzen were fitted on all StuG IVs throughout their production run so that shouldn't be cause for concern.
HTH!