As such there is no need to add your aircraft to the flying school's AOC, nor does the school need to see anything other than the current maintenance release.
The aeroplane would be
operated under the AOC and the flying school assumes the responsibilities of the operator. Any decent school would need to see a lot more than the MR. The school has a duty to the instructor as well as the student, and would certainly not be meeting their duty by assigning their instructors to an unknown aeroplane. Any school that would is unlikely to have decent instructors in the first place...
The maintenance release can even be classed private, it doesn't even need to be Airwork, that only becomes an issue if you want to hire it out to the school.
Rubbish - the flying training described by the OP is aerial work.
If you are teaching someone in their own aeroplane, however, you are becoming the defacto aircraft operator and most schools will have a list of things to check and do to bring the aircraft in line with their existing fleet, paperwork, etc.
and
Besides insurance, the other major issue is that the training organisation's HAAMC has to become responsible for he aircraft's maintenance. AD's, SB's etc. have to be ensured as being carried out.
Exactly right