dscoll, there is an option to tile a smaller area, and discard any tiles damaged as they are being cleaned up to re-tile. For example, you mention that tiles are not full ceiling height all the way round - drop the height of the tiles by one row or two.
I accept you cannot afford to re-tile, so if money is a problem, use your own time to rectify the problem. And yes it will be a long and frustrating piece of work.
The tiler employed cannot be a pro if he is fixing wall tiles using 10mm thickness of adhesive. This does not help the tiles stay on the wall.
If sticking tiles on a wall with pva was a serious option - then we would all be using pva instead of the tile adhesive.
Accept this as a very serious exercise in learning to find the right tradesman to do the right job.
Finally a word on your friendly builder. It is great to see him accepting some responsibility and offering to rectify the problem caused by the tiler. However, would you ask him to fit a new gas pipe or re-wire your consumer unit? If the answer is no, then consider that the tilers professional advice might be right, even though they contradict the builders opinion/suggestion.