mm43;
If the APU had been started, would the Cabin Pressure controller have been active? I haven't seen any data on this, and have assumed that should there be no electrical supply from the prime sources that the controller would operate below 8000/7350 feet to equalize the cabin pressure.
The #1 CPC is powered by the DC ESSential bus, the #2 CPC is powered by the DC1 bus and CPC Manual Control is provided by the DC BATtery bus.
So yes, the control of cabin pressure would have been fully available with APU and even with with no APU with manual control from the DC BAT bus.
Chris Scott is correct regarding negative pressure relief; the single valve is installed aft of the L1 door.
Takata;
Just for info....it is not possible to say without very detailed examinations and perhaps simulator investigations, that the CPCs would be affected by loss of just the 3 ADRs due to iced pitots only. I wondered about this but cannot determine it from the FCOM.
Some data may still be provided to the CPCs or it may not - the FCOM is very simplified and geared towards the flight crew "need-to-know" level. It is not entirely possible to confirm the effects of loss of just the static ports either, without a similar investigation.