Hi Wyatt,
The 8.8cm Flak on Sdkfz 9 was incredibly rare (only 15 built) and was probably gone by the latter half of the war. All 8.8cm Flak that I've found for 1944-45 was towed.
8.8cm Flak Batteries typically had four weapons (some had six), plus a couple of 2cm Flak for point defence against jabos.
Most late-war Flak battalions had two or three 8.8cm batteries and a single battery of lighter SP guns – typically a mixture of Sdkfz 7/1 (3.7cm) and Sdkfz 10/4 (2cm). alternatively, these might be towed 3.7cm guns.
As mentioned, all divisional Flak battalions (except for FJ Divs, Pz Div Hermann Goering & LW Field Divs) were Army or SS. However, note that all Luftwaffe Field Divisions were transferred to the Army in November 1943 (though uniforms took a while to change over – a mixture was photographed in Normandy).
Corps and Army-level Flak battalions were relatively scarce, though they did exist and were managed by the Army. Most rear-area Flak defence was managed by the Luftwaffe Flak-Korps, though these frequently found themselves shoved into the front line, such as III. Flak-Korps in Normandy.
The Flakpanzers (FlakPz 38(t), Moebelwagen, Wirbelwind and Ostwind) were operated exclusively by the Panzer Regiments and were organised as described by Gary above.
Panzer Battalions, including independent Tiger battalions, also tended to operate a single platoon of three Sdkfz 7/2 (Quad 2cm).
Sdkfz 251/17 with 2cm Flak tended to be used as panzergrenadier platoon commanders' vehicles very late in the war, filling the role once occupied by Sdkfz 251/10. However, the Luftwaffe version of the 251/17 (with drop-down sides) I believed was employed by Flak battalions.
As Gary says, Sdkfz 251/21 was generally only used by the independent Panzer Brigades.
Grenadier Regiments could sometimes have a regimental Flak Company consisting of towed and/or SP 2cm and/or 3.7cm guns, though they were frequently massed into divisional flak battalions or into one of the companies of the divisional panzerjaeger battalion.
Panzergrenadier flak elements are described by Gary above.
The general rule of course, is that there was a bewildering array or organisations and equipment and there were plenty of exceptions to the rules. For example, after Normandy, the three surviving Moebelwagen flakpanzers belonging to SS-Panzer-Regiment 9 were taken away and given to SS-Flak-Battalion 9 (with whom they fought at Arnhem).
Remember when painting that Flak waffenfarbe would only be worn if they belonged to a dedicated flak battalion. Flak units belonging to Grenadiers, Panzers, Panzergrenadiers, etc, would wear the waffenfarbe of that branch, not the flak branch.