1. if they blink for the security then your bulbs and fuses should be fine. I would check the switches and relays.
2. Could be a blown fuse, bad regulator, bad switch, broken control arm, worn motor, or combination of any.
3. If the alarm goes off when the car is armed and you inset the key then it is functioning correctly. It is sensing the motion and vibration on/in the door and activates.
<span style='color:gray'><span style='font-family:Courier'>1995 e36 3 Series RIP
1999 e46 3 Series BURN IN HELL
2000 MR-S Roadster= Current</span></span>