Old cable: type of isolation and size of ground
That cable is old cloth jacketed nonmetallic cable.  I would say by the 1970's, this was no longer being installed, nonmetallic cable in the 1970's had a plastic jacket not too much different than what's used today.  

The ground wire was considerably smaller on the old cloth jacketed cable, and the early plastic jacketed cable.  I see a lot of old 12-2 romex, which was white back then rather than yellow typical of 12-2 now, with a 16 gauge ground wire.  The smaller gauge of the ground wire isn't usually an issue because it's usually not normally carrying current, it's intended to just carry fault current briefly so the breaker trips (or back then, the fuse blew).  In some situations, the fault may not clear, and it's conceivable in those situations the wire could become overloaded, but that's unusual enough I don't think it's considered a major concern.  

If the insulation is undisturbed, it's usually fine, but if it has deteriorated with age, it can be brittle and crumbly, so it's best left alone.  It's especially prone to deterioration if it's been subjected to a lot of heat - in attics, above light fixtures, etc.