All of the CTs in the store have TVS diodes that will limit the voltage that they can produce to safe levels when disconnected. That said, belt and suspenders, I do recommend physically shorting them If unplugged for more than a short time. Wrapping a strip of aluminum foil tightly around the jack work.
I would not use solid core unless it were a space issue. Accuracy is a purely academic consideration. As a practical matter there is no other advantage (except I suppose slightly lower cost).