CRJ drivers and those really "in the know" about this accident correct me if I am wrong, but:
The engines would not have relit during a properly executed APU start because the cores were already cooked. This was potentially due to the crew trying to windmill start the motors without enough airspeed (and thus N2 rotation) earlier in the accident chain.
Again, that is not correct. The cores were not "cooked." Only one engine suffered heat damage, and it was not enough damage to prevent a start. The engine sustained no heat damage whatsoever. This is all available in the NTSB docket.