Oliver2002 has got the horror story right. To add PIC was a senior TRI and P2 was just starting his SLF (Supervised line flying) in Ai parlance.
A lot of holes in the Emmental cheese threat matrix with training, APU inop and a new flight crewmember.
A quick question. Is setting the flaps an after start item?
One operator I often work with sets the flaps after engine start but before the pushback is complete. On some of their a/c this puts the engines in to flight idle. An increase in engine power while the headset man is still around. I have tried to determine the logic of this as not all of the a/c do it.
I've seen several flap setting strategies in airliners over the years. The early jets like the 707 and DC-8 had low wings and the fear was that a fire bottle or something else left on the ramp might ding the flap (and somehow miss the engine) on taxi out so the flaps were not extended until clearing the ramp area. It was also a common 'technique' to disable the takeoff config warning horn ('set off by the white knobs' on the Boeing oral gouge) to avoid spurious alerts while taxiing so the situation was set up for a calamity.
I flew with a couple of operators who reset the flaps to a takeoff position after landing thinking that would default to a flyable configuration for the next crew if all else failed.
These days we set the flaps when the aircraft first moves forward under its own power. Unless there is frozen precip, contaminated taxiways and such, in which case we taxi out with flaps up but the PIC has to devise a 'plan of action' to ensure that the flaps are properly extended for takeoff. The Styrofoam coffee cup placed on the flap handle is one popular visible reminder that something needs to be done with that handle before takeoff.
As you observe, some planes go to flight idle when the flaps are extended on the ground, others only do it only when the flaps are extended while airborne. Also, turning on engine heat for taxi, which would be on the after start checklist, also bumps up the idle with some motors.