APU supplies power when on the ground and not plugged into the ground supply ( 28VDC @8kA peak and 115VAC 400Hz @50A) so that the systems can be started. The APU is what is used to start the engines, it is a little dicey to use the main battery to do that, as it is capable of it ( absolutely, it will deliver 8kA into a load with a terminal voltage of 22V) but then the charging will need limiting to prevent overcharge. These batteries have a slight negative temperature coefficient and thus droop in voltage as they get hot. Can be fun if you have a starter generator on the turbine which is quite capable of charging at over 2kA, and a simple carbon pile regulator that lowers current as the voltage increases. Newer aircraft just limit the charge current to something the battery will accept as overcharge without issue.
Batteries are often only monitored for voltage, and this is from a sensor set on the main DC battery bus. There are temperature sensors on many, but they are only going to tell you that the pack has exceeded 130C case temperature. ie it is either going to go or is actually cooking up.
I do not know of these Liion packs, but I would surmise they are only monitored by the maintenance processor and are not actually connected to the main warning system aside from the case sensor ( monitoring being done on DCbus1, DCbus2, DCbusaux and Dcbusemerg for voltage only) which may not be used ( half the planes I worked on either did not have the sensor in the battery, and a socket next to the battery to plug it so the warning system would not show disconnected; while the others did not have the wiring at all.) on all of the planes.
Ram air provides hydraulic power only to some flight systems, no electrics, that is done by the battery only, and will only be powering the standby artificial horizon, the standby radio and a set of emergency lights in the cabin and cockpit. All other systems are gone aside from the minimum - attitude, altitude, compass heading and basic controls. landing gear is not supplied, you have to manually pump the emergency release and pump the doors open and then drop the gear by gravity and pump it till it locks. No brake power assist or ABS either, you climb on the pedals with both feet and push hard, and hope you stop before you run out of runway or that the tyres do not blow out from the heat.
Only one time that I was wishing I had a parachute when flying. But we landed and were able to walk away, and the plane was able to fly again as well.