I run the MF31-1000 in my P38, and have done for the last 4 years. It's supposedly a Lead/Calcium battery - but seems to charge OK on the p38 system (just make sure all the power/ground wires are all up to scratch!)
On a few of the other points...
EAS inhibit - the BECM combines all the 'door ajar' switch signals, and the 'tailgate open' switch and sends that down one wire to the EAS ECU as a 'door open' inhibit line. This is a ground signal when any of the doors or tailgate are opened, otherwise it should sit at battery voltage. It runs from the BECM on C112 (16 way green connector on the front of the BECM) pin 3 - Purple/Orange wire over to the a splice which splits it to both the EAS ECU and the EAS delay/timer relay module. It could be worth looking under the passenger seat at the wiring to see if you can find the splice, or see if the wiring has been cut or damaged. If it's somehow become disconnected, then it will give the symptom you show as it will never pull the wire to ground. It could also be worth checking the pins on the green connector at the BECM. My mate had the problem on his P38 and the issue was found at the green connector where the pin had backed out of the connector a bit, so wasn't making proper connection.
As Wammers says - there is no option to turn it on or off in Nanocom or any other diagnostics.
Door locking etc - It sounds like the motors in the door latches have burnt out or otherwise failed. The L/H/R one might be a wiring thing as it runs directly from the BECM (though this is not likely to be the problem as from memory both rear doors are triggered from the same relay). The front doors... more than likely the motors there too. The fact that you can open the tailgate says that the Central Locking switch in the drivers latch is working - as the tailgate motor grounds through there. If the doors/mirrors all work in the front doors then the communications from the BECM to the front door outstations are all working ok - so it points to the latches being the culprit.
I refurbish door latches in my spare time and see a LOT (mostly fronts) where the motors have burnt out.
Hope this helps,
Marty