What a pain in the ersa! I love working in scruffy houses when the people themselves are the type that wouldn't notice let alone care if you do cause cosmetic damage because the house is already scruffy (because they generally are more interesting to chat to), but it sounds like you've found a sisp-taker (who may genuinely, however, have only just noted a slight deterioration in a carpet that was already worn).
The 'damage' may be hard to prove or disprove, though it's up to you whether you want to go for the easy life or call their bluff and see if the customer will actually go to court. If you didn't take notes and have no evidence of the condition prior to work taking place, I think it may go against you, though if you want to fight it, you could offer a percentage based on the condition of the carpet which you have not applied your protector to being already very poor. Perhaps you could offer 33% of the cost of a new carpet as the amount that is not in dispute and give that upfront? At least you would probably then have the judge's sympathy. This, incidentally, is very similar to best practice method if it were a rental property and the carpet had been 'damaged' by tenants (cost of new carpet x percentage of the expected life of the carpet remaining).
As for the shaver socket, did the customer authorise the work or not? If the customer authorised it and didn't agree a price, they can be expected to pay the reasonable cost. I think it would be hard to argue that £75 is unreasonable.
The electrician needs to understand the system and determine whether the circuit is fit to connect a new socket to, do the work and clean up after (which may indeed take only an hour) and then test the new socket in accordance with BSblabla. The total labour cost is the time spent on the job, not just the time spent chiselling out the wall and tightening the terminals.
Plus, if the sparky is under your subcontract, then you need to add a small percentage for the administrational cost and responsibility you take and that you incur in providing the customer with the convenience of having you as the single point of contact, same as you would put a small margin on the materials you supply. FWIW, back in 2008, a Colchester-based electrical firm's callout and first hour's labour charge was significantly more than the £75 which your customer is disputing.