1. There is nothing wrong with the watch. A simple workaround solves the problem.
2. There is no independent setting for DST as you describe for the 5121 module. If one sets the watch for a time zone that receives a signal, DST is ALWAYS set to AUTO. DST can temporarily be set to ON/OFF, but this setting will be overridden when the watch next syncs successfully to conform to the DST info in the time signal.
3. Setting the time zone one zone back will have no effect on world times. In this case, setting the watch to LAX (GMT -8) will display the correct time for Arizona when LAX is on DST (GMT -7). Since the watch thinks it is in LAX and displaying the correct time for LAX, all other world times will also be correctly displayed. The fact that the watch is physically in Arizona (also GMT -7) has no effect on world times.
The only way to turn DST permanently ON or OFF is to set the watch for a time zone that does not receive a signal and set all time/date functions manually. In this case, the watch will not even attempt a sync. If the watch is set to a time zone that receives a signal, it will ALWAYS be overridden by the DST information in the time signal.
HTH