The doorlock problem is not related to the alarm system. That is simply a problem of one of the microswitches or another problem in the locking system of the doors or the trunk lock.
See here under body, doorlock etc
http://bmwe32.masscom.net/
The system is , that the GM reacts, as a signal is missing that all doors and trunk are unlocked, so it locks again.
You can also try resynchronizing the locks. See owners manual.
The problem with the no start: you sais you changed the battery. Was there a small microswitch on top of the battery? If so, maybe this one is not correctly installed. It is part of the drive away protection system.