I'm sure the electronics had a regulator, probably to 12 volts, but possibly to 5. I expect the raw supply was used for relays and
solenoids, when present. The battery pack would deliver 14.4 volts from rechargeable ni-cad batteries.
Your understanding of the single-sweep function is correct.
The C-5 (and C-12) are older cameras, designed for the 500-series 'scopes, some of which did not have illuminated graticules.
A flash could have been used for those, but it would be a very special one since the graticules required edge-lighting, not face-on.
I've never seen such an accessory, but it's quite possible. All my 500-series 'scopes had illumination.
The C-31 cameras were designed for the 7000-series 'scopes, much later vintage and all had graticule illumination.
Have fun with the toy.