I can only reply re the 777 but a quick overview
As stated, the apu will autostart in the event loss of both AC Buses, but before that,
If an engine is sensed to be below the normal idle for the prevailing conditions, the the EEC,s will try to recover it, simply by starting the ignitors. As a result of this, RR/GE powered 777 do not have a selectable cont ign setting.(it may be an option for the GE,s)
If this intitial action is unsuccessful and the N2 goes below 35% then the engine is considered to have failed and displays as such.
All this happens without the involvement of the autostart.
Can result in a bit of confusion, can get an engine fail warning, but the engine is recovered, and operates normally.
Subsequently, the system will access the operating enviroment and indicate if a cross bleed start is advisable. This does not mean that a windmill start will not work, just that the best chance is with cross bleed. The system seams to prefer windmill starts if it can, prob because they are easier???
Once the start sequence has been manaully initiated, then the autostart will continue to try until it is told not to by manually turning the start off. This may be the airbourne autostarts you are talking about.
If both engines fail, then the sequence is basically the same, but with a lot more bells and warnings and higher pulse rates.