Remember these ADP's are demand pumps and only cut in when the EDP can not supply enough pressure, normally in high work load situations, gear/flaps etc etc.
When selected to AUTO, yes. When selected to CONTINUOUS, no.
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!)?
The batteries are already online, and in absence of generator power, external power, or APU generator power, are the sole source of electricity.
At this stage one is down to standby power, with few items powered off the standby bus, and a short time to enjoy it.
EDP's are engine-driven hydraulic pumps, and don't power anything with electricity, as they only pump hydraulic fluid.
AC motors, or the AC hydraulic pumps, are powered by electricity...but have nothing to do with the engine driven pumps. The electric hydraulic pumps, one on number 4 system and sometimes one on number 2, provide supplementary hydraulic pressure for ground operations, such as towing and powering body gear steering (hydraulic system No. 1) and for applying the parking brake (hyd system No. 4).