The door lock works whether the engine is running or not, if doors are open it will attempt to lock, and sense one lock has not engaged, depending on type of lock(some will) cossies's wont and then immediately unlock.
So yes start with one fob on keys lock with another.
There is a possibility of AUTOLOCK even when a door HAS been opened, thius is due to dirty contact switches, ie it doesn't sense a door has been opened so you start car open door to get out 10 seconds after starting autolock kicks in, so make sure none of the door switches are ropey.
The autolock can be disabled by going through the KEY positions with the FOUR digit code you got with the alarm.
You can also enable or disable the BEEP on arm/disarm or alter the beep volume to one of 3 levels, but even quiet level is loud.
They should come with beep on arm/disarm disabled as it is illegal in rest of europe due to noise pollution.
tabetha