First of all, which wire is ground in an analog video cable? Is it the inner wire, or the outer 'shielding' wire?
Always the outer braid... it's like a Faraday cage.
Secondly, can I share that ground wire with the ground from a 12v source and still have clear video?
but in general the shielding should only be grounded at one end to avoid forming an earth loop.
I don't think so, not for single-ended lines.
Now, if this is a balanced/differential line, then you have the HOT and COLD lines, and you have the shield/ground. Then you can connect your shield (pin1 usually on XLR or TRS jacks) grounded to the case immediately at the entry point (use the metal tabs on the XLR jacks provided for this - short pin1 and the metal tab with a glob of solder, then secure XLR jack to metal case with screws.)