Alarm going off randomly 2007 Fit LX
 

I just picked up a second hand 2007 Honda Fix LX and the alarm seems crazy. Some episodes:

1. If I don't lock the car and then open a door
2. If I don't lock the car and don't do anything
3. If I lock the car and lift the driver's handle
4. No start despite no alarm going off (lights and fuel pump come on). Rectifies automatically after a few tries
5. 0.5 - 1 second delay to starting from turning the key

Any idea what could be causing these problems?