Hey Shrew, this is not the first time I have heard this on how the alarm is "supposed" to operate. I can tell you my 1996 5.7 Alpha 1 Gen II does not work this way.
When I turn on my key the alarm sounds and does not turn off until engine is started (I have waited 3 minutes before getting annoyed with alarm sound). I have heard it should shut off after 30 seconds or so, maybe mine is broke
Now I have never overheated so cannot attest to that one.
I did blow my front shaft seal earlier this year and the oil monitor went dry. After I started the engine the alarm never went off and just stayed on until I turned off the engine. Again, maybe mine is broke but this is how it works on my 96 model