Respecting that this thread is titled Propeller striking....
It seems however to be focused more on the "Over voltage" warning light, as a contributing factor... Referring to the C150/152/172 system (but I'm sure that the Tomahawk is very similar), I do not agree with:
...... It should go off when sitting at 1200rpm.
It means that there isn't 14V in the electrics which means the battery isn't charging from memory it triggers under 13Volts. Which in the grand scale of things for PPL is a very good reason to shut down and get another plane.
The alternator can have some variability in it's low speed (RPM) efficiency. If the Low voltage light is
not indicating at high idle to flying power engine speeds, and your ammeter is not showing a discharge, the plane is fine. If it is indicating at mid range RPM, you should expect to see a discharge on the ammeter also (though a small discharge can be hard to distinguish). Landing light on, and increase RPM (during run up is a good time for this). If you now see a discharge, then take the plane back, you do have a fault. Try cycling the alternator side of the master switch if it is split.
If in flight, you have an alternator failure, (ammeter discharge indicated, LV light on), with some economizing of electrical load, and a decent battery, you have a half hour of flying with no problem.
You will not hurt the engine running with the LV warning light on - for any length of time - it's approved to run that way. (many of these engines have direct ancestors who were in aircraft with no electrical systems). You just won't have battery charging going on. I have not seen oil pressure warning lights on these aircraft, but if you have an illuminated low oil pressure warning (some STC'd instruments have this function), then yes, on the ground, shut down with no delay. In flight, plan for an engine failure soon, unless you're content to shut it down and forced approach from where you are.
Few aircraft have "starter lights" but they are a great idea. If you have one, and it is warning you of a stuck starter contactor, shut down right away.
As for not shutting down right after a start, no problem. If you would like to leave it running for any appropriate reason, keep non pilot passengers inside with you. ANY time I allow a pilot passenger to exit with the engine running, I'll turn the aircraft so their exit path is away from the turning prop, and still remind them about the running propeller.
Passenger safety is
totally the pilot's responsibility!