Can a Support class soldier have a total of 8 medikit uses?

Short answer: No

There is a 1-medikit limit per soldier (thanks ken). Medikits, chitin plating, mind shields, nano-fiber vests, scopes, reaper rounds, and respirator implants are all limited to one-per-soldier.

Those restrictions were present in EU, too, when Deep Pockets meant only the Supports could have two items - and back then, it was a single 3-heal medikit and any other item.

However, I can confirm that Training Roulette can be fun - Deep Pockets stacks with Grenadier (each pile of grenades is 3 deep). And you can equip two grenades, for a total of 6, each doing extra damage.

Final answer: Deep Pockets stacks with Field Medic (4 uses per medikit), but that's it. No synergy with Tactical Rigging.