How and where does the ADP get it's source of air from? Directly from the engine bleed air?
In the air, engine bleed is the only source of air... unless you left the APU running after starting it on the ground (and were below a certain altitude... the APU runs out of "steam" at higher altitudes)
Should all the 4 engines decide to quit would there still be a source of electrical power without having the batteries to come online since the AC motors are powered by EDPs (if I remember reading it this way!)?
To get any electric powered hydraulic pumps to operate in the air, you need the engines to windmill fast enough so that the high speed rotor (N2 or N3) is above 50%. Engine electrical generators do not operate below this speed.
Electric pumps on #2 and #3 maybe be considered optional.. but there is a lot of aircraft flying around with this option. I wouldn't be surprised if the electric versions outnumber the non-electric ones.
Note that you still need main bus electrics to control the air-driven pumps, so, again, windmilling speed is a factor.
Rgds.
NSEU