Split-Core Current Transformer - Disconnecting secondary load questions
The voltage spike is caused by the secondary coil not having a discharge path: Inductors oppose change in current flowing through them. Disconnecting the coil forces the current flow to stop instantly. Thus the inductor must find a path to discharge the energy stored in it, and unless it finds such a path, the behavior is of voltage across the coil rising to an arbitrary level. This mechanism is very effectively used in a variety of designs requiring very high voltages, such as some automotive ignition coils and other such devices.
Keep the load resistor in place. Disconnect the sensing connection alone.
In the secondary.
Yes, the burden or load resistor is designed to provide a current path, with the voltage read across it. The sensing device's impedance appears in parallel to the burden resistor, splitting the coil current. If the sensing equipment is high impedance, e.g. a solid state multimeter as opposed to a low-impedance vintage coil voltmeter, this parallel load is insignificant. 
No.