Why does the bell ring when I put the key in the ignition with all doors shut and engine running or off?

The bell will sound whenever a door switch is broken or its wiring is disconnected. Sometimes the door switches (which are located in the front of the door jamb, near the hinges) can fall inside because the plastic tab physically breaks off and then the system "thinks" the doors are open.

If the switches look secure, remove them from the doors, and with a voltmeter, check for a state change when pushing in on the switch with the wiring disconnected. No change, bad switch. If all four check OK, then the wiring to the door switches should be checked. The two primary connectors are underneath the driver's side bolster and passenger side kick panel. The lines terminate and connect here.