It sounds like there's an ungrounded (hot) conductor shorted to ground somewhere. Finding the short is likely going to be a tedious task.
Check your work
Did you install grommets or clamps, where the wiring passed through knockout holes in boxes? If not, start by inspecting the wiring where it passes through the knockouts. The edges of these holes (especially on thin recessed lighting boxes) can be quite sharp, and can easily damage cables and wires as they pass through the hole.
Check all your connections, making sure nothing came into contact with ground or non-current carrying metal. This can sometimes happen when wires are pushed into boxes, so check for loose or pulled out wires.
Inspect cables/wires where they are supported, especially where supported by metal staples or clips.
Check existing work
Start by turning off each breaker in the panel one at a time, while monitoring the voltage on the drop ceiling rails with your non-contact voltage tester. When the voltage goes away, you've found the offending circuit.
Once you know which circuit is causing the problem, it's time to start the tedious task of locating the fault. Do your best to trace out the circuit, and determine all the equipment on it. Open up all outlets and equipment, and inspect the wiring inside.
It may help to move along the circuit, and disconnect the load conductors (the conductors feeding the next junction) at each junction. Then energize the circuit, and check for the voltage again. This can help you isolate the problem, and will help you focus on the problem wiring. Once you find the offending part of the circuit, methodically inspect the wiring. Checking for damaged wiring, loose connections, improper wiring, etc.
Unless you get really lucky, you're likely in for a long tedious search. Good luck.
Yes, this rod is correct, only it is NOT "additional", it is required since this feeder runs to a detached structure.
Also, as mentioned, many/most inspectors will as if the ground rod meet the 25 ohm requirement. If it does not then a second rod is required and the 25 ohm rule is no longer in place.
Also, and this is a biggie, that SOOW cord is absolutely NOT code legal in this case. Rubber cord CANNOT be used for permanent wiring. You need to use either individual THHN/THWN conductors if the conduit is complete from panel to panel, or you can use UF cable as long as the size cable you get can legally fit in the conduit.
Individual conductors in conduit could be #8cu, but due to the distance and the small cost difference I'd go with #6, actually #6 for the hots, #8 for the neutral, and #10 for the ground.
Uf cable would need to be 6/3.
I would seriously consider making this right and getting the whole job inspected.
Best Answer
Keep in mind that you can't use wires pulled out of cable because the insulation is not marked. You also have to have all splices inside a junction box.
But to answer the question, you can't use a red wire for ground, but you can strip all the insulation off the conductor because a bare wire for ground is OK. It might take a few minutes depending on the length and the tools at hand.