Try taking it to an empty parking lot where the noise won't bother anybody, then test the doors and see if they set off the alarm. Close the door, arm the alarm, then use the key to open the door without disarming the alarm. It should go off (I doubt that the alarm senses the door-lock position). If it doesn't, then 8's probably right. Your door switch is broken and needs to be replaced.
Come to think of it, I should probably test mine. I'm kinda' just taking it on faith that it will work if needed.
Also, the "magic" behavior from the siren makes sense. I haven't checked, but I imagine that the alarm siren may be grounded through the mounting bolt. It probably has only one wire going to it (providing power - positive voltage - when the system wants to the alarm to sound). However, current won't flow correctly (and thus the alarm won't chirp) unless the negative side (ground) connection is also made. When you tightened the bolt, you restored ground and fixed the problem.
Hope that helps,