WRT-The issue of APU running for ETOPS.
The B737 when operated on an ETOPS sector needs the redundancy provided by the APU to cover generators. Because the APU will not (reliably) light above 25000', the APU is left on after start, and remains on throughout the flight until the end of the last ETOPS segment.
So I would imagine it is aircraft/fleet specific.
I cannot obviously comment on other types.