I think it's as gazza says. The contacts somewhere under the slow speed relay are bad. Aircon, overheating and a disconnected sensor all (initially) call for slow speed, so the relay closes but nothing happens. Shorting the sensor contacts should call for high speed, which should work. It's possible though that the ecu will see this as a fault, and just try to engage slow speed again!
Reconnecting the sensor turns the slow speed relay off, hence the tick. If the engine was overly hot at that point the sensor is under reading. Otherwise it is probably good.
The ecu probably checks the relay coils. When you unplug a relay the ecu detects the open circuit. When you replug it the ecu sees the coil and runs the fan for two seconds as a check (maybe...).