I've met a few PFEs, FEs and FOs who aren't worth their weight in
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I've met many who are average, and I've met very few who are fantastic at what they do, PFE or FE (or FO) doesn't matter, the smart ones stand out regardless of their background.
Also, I'm not sure about your particular aircraft or the way it was equipped, but if the bell rang right at rotation it may be that the APU was wired in with the WOW switch to generate a bell if the APU was on at liftoff, on the other hand you mentioned the light being on as well, once those detectors get heated to a certain level the fire may well be out (or the heat from the APU) but the light won't extinguish for a while. I know in some 727s the APU being on generates a T/O warning horn and on others it does not. The switch shouldn't be disabled in flight, the APU fuel valve is disabled from opening, but it is not disabled from closing in flight. So you can leave it on through takeoff and turn it off in flight (like I once did), but you cannot turn it on in flight.
Also, funny how they taught you to look for the WW fire as the first indication of APU being on, but you got the APU indicating a fire. As I have told many people flying the old birds is a lot more black magic and know how than it is memorizing what to do when what light comes on. Over the production run of the 727 there have been many changes as to what happens when up front but not that many in terms of the systems in the back, this leads to one 727 giving one certain indication of a problem and another 727 giving a completley different indication for the same problem.
But it does sound like your FE wasn't willing to own up to his mistake, never heard of a self generated start and don't think one is possible with the way the APU is set up in the 727. Not being able to own up like that is one of the biggest red flags for me in terms of a crewmember.