Electrical – Ground wire in 14/2 NM-B cable is at 60+ Volts when not connected to anything

What you are describing is called a Multi-wire Branch Circuit, and it is completely legal (given that it's done properly).

The first problem, is that it doesn't should like the breakers feeding this circuit are handle tied together. It should be feed by a double pole breaker, or the two breakers handles should be tied together. You should not be able to shut off only half the circuit (red only or black only).

As for the voltage. I'd suspect it's phantom voltage (inductive coupling), and so should not cause any problems (especially if the breakers were properly tied together).

If the breaker for the "red" conductor is off, and current is flowing on the "black" conductor. Because the conductors are parallel, and in close proximity, a voltage may be induced on the "red" wire. There should be no current, however. So if a load was introduced, the voltage would drop to 0.

You can determine if it really is phantom voltage, simply by using a low impedance voltmeter. The low impedance meter will read 0 volts, while a high impedance meter will read the phantom voltage. An analog meter is one example of a low impedance meter, though some digital multimeters are able to operate in both modes.

From your photo, it's clear the electrician left you a handy little gift -- those third white and bare wires are pre-installed pigtails. With the power to the circuit off of course, simply strip the white pigtail and the unstripped black wire and hook up your receptacle (free black wire to brass screw, white pigtail wire to silver screw, bare pigtail to green screw).