What the controller sees depends to a large extent on what radar processing system is used.
What happens often depends on code callsign conversion.
Taking the 7700 case first. If one sets 7700 then an alarm goes off etc etc. However since many radar systems are programmed to know that 7700 is an aircraft in distress then when the 7700 disappears, the computer will freeze the target at a location so that SAR aircraft can be directed to that place. Thus the busy controller has to take time out to clear the false alarm.
On the normal codes, if the processing system has been given your callsign then it will display your callsign insted of the code. If the code changes or you select standby, then the computer will do a thing called coasting the target. This means that it will continue to paint the target a positions where it expects you to be. The symbol will change slightly to indicate this to the controller. After about 3 updates, the callsign will disappear. The reason for this is that not every transponder reply is correctly received and processed by the ATC system and thus it makes allowances for mised replies.
If someone else selects your code then the computer will remove your identification from the target because it knows that ther can not be two aircraft with the same callsign.....better to have no aircraft displayed than the wrong one.
I expect that in the USA, the continuity of the system ensures that every unit has your code and callsign programmed in advance so that the target with your name attached will simply coast whilr you fiddle with the codes......provided you don't select the 75,76,7700 codes which will stop the coast and set off the alarm.
regards,
DFC