yes you will be able to hear the relay click, it is over on the passenger A pillar near the door sill.
while it's not normal, it is possible that the alarm kills power to the computer. But even so the alarm would not immobilise immediately that you turn off the car, so it won't be that.
I just checked on the stagea, the PFC hand controller turns off when the eccs relay does, about 1 sec after turning off the ignition. BTW the eccs relay un-earths the computer, it doesn't cut the power.