I ampretty sure the "series of clicks" after you lock your door would be a problem with your door's actuator. This is the thing that locks/unlocks your doors locks. This happen to my accrd once.

The locking after you start the car seems to be an aftermakret alarm ssystem. They can be programed to lock when your key is in the ON position. But since you say you dont have one, theres probably a short somewhere like everyone else said.