Interesting. Someone else has a 126 that has a passenger side key mechanism that will unlock but not lock the doors. If you don't mind, can you tell me if you can press down the inside lock button on the passenger door when the door is open?
The factory alarm is set by locking the driver's door with the key. If your alarm bothers you and you don't need it, disable the alarm. On my car, and yours is probably similar, lift the front passenger carpet, remove the false firewall, locate the alarm box marked 'Becker Autosound,' remove the connector furthest from the floor. Leave the other one in place. On my car and one other member's, this disables the alarm without disturbing anything else.
Thanks and good luck,