If I can remember my traction bumpf, I think you're pretty much spot on
The a/c is powered both by the ETS directly (fans & equipment) and by the coach batteries (control systems), which in turn are charged continuously by the ETS supply. The lighting also comes off the batteries, as do the control systems for door operation and buzzers etc. As you say, catering is ETS directly, as are the plug sockets in the luggage racks (but the at-seat is battery). In normal operation the rear power car provides the supply, if it fails for any reason the Driver will switch to the front. The voltage actually varies depending on the engine speed, but 415 is the maximum output, 270 being the minimum. Unless both power cars or their ETS supplies fail, there will always be a power supply to the a/c and the system should be 'on' (unless it has been isolated for maintenance reasons of course).
In most cases there will be coaches within a set which are suitably cool, and others which aren't. Whilst the electrical supply is common throughout the train, obviously the a/c equipment is individual per coach and so will vary in condition and operation. Coach D tends to be warmer to begin with as it is full of televisions, which do generate a fair bit of heat between them. Obviously the area around the buffet kitchen can also be particularly toasty. I've seen defect sheets where Fleet have suspected a simple lack of coolant gas, and others where major a/c components have failed. It's a common problem, and of course the need to have working air conditioning has to be balanced against the need to have trains out in service when the fleet allocation is permanently tight.
It does seem in general that the specification for air conditioning systems supplied to the UK needs to be re-examined, as failure even of fairly new kit is far too widespread.