Play in the diff! Thats not part of an MOT test! Transmission wise all that is required on the MOT is for the car to move! There could be an MOT issue if the gearbox is leaking oil excessively but otherwise its not part of the test.
The only transmission issues concerning the MOT are with the driveshafts....both the inner and outer rubber boots mustn't be split.
Diff bearings aren't tested.
Its probably worth looking for a replacement box rather than rebuilding your old one - financially anyway - but I'd be seeking a second opinion here.
John
Has an engine mounting failed, so the whole engine / gearbox / differential assembly can move about excessively, risking (in the eyes of the examiner) a drive shaft falling out?
Yes they do and yes they could! The only reason an outer split CV gaitor is an MOT fail is because grease could find its way onto the brakes, as far as I know the inner gaiter is not part of the test.
You are absolutely correct....from the testers bible -
'Steering rack gaiters and front outer constant velocity joint boots are examined'.
However....'The condition of front drive shafts are checked as are the condition of CV joints if applicable'.
Considering the cost of driveshafts, its a good idea to replace the boots if they split!
John