On ATC's radar, does an aircraft that loses its transponder simply disappear?
It depends on the system. Assuming only secondary radar is available, in some systems, the track will indeed just disappear when radar contact is lost. Other systems will put the track in "coasted" mode, continuing to show the track (but with another track symbol), waiting to pick up the transponder again. The track will then be terminated after a specified time limit.

Of course, if primary radar coverage is available - which works independently of aircraft equipment, such as transponders - the system would still be able to track the aircraft, but automatic flightplan correlation would most probably be lost, along with mode C data (altitude). So all flight information would vanish from the screen and the controller would have to re-identify the flight using proper procedures.

Producing an alert when a track is lost could be useful, but it could also potentially result in a lot of nuisance alerts, particularly in areas with spotty radar coverage. So alerts should only be shown for areas and levels where radar coverage is expected to be perfect, otherwise the controllers would likely develop a habit of ignoring such alerts. Also remember that radars lose contact with flights all the time - as they move out of the range of the radar. No ATC station has global radar coverage. Showing an alert every time an aircraft leaves the area of radar coverage would also be pointless, and the range of a radar can vary from day to day depending on atmospheric conditions, so defining a precise area in which alerts should be suppressed might not be a trivial task.