So the new electrician has just been. Apparently the plug sockets for the dishwasher, washer and dryer were the wrong sort: they need to be specifically for large appliances, rather than just a "normal" one. This was clearly the landlord taking a shortcut. (He did a lot of the electrics himself.) The electrician replaced one plug socket and will come back to do the others. He affirmed that it had nothing to do with my toaster and blender and in fact it was okay to use extension cords in the kitchen!

He was not bothered about the plug under the sink at all. He said there was no water damage, just damage from the wrong socket type.

For the rest of the electrics, he said that appliances had been added to circuits incorrectly (there are three large appliances on each circuit instead of just one) and recommended redoing the whole thing, although there is no fire risk. Not sure if the landlords will be keen but I'm just relieved that the problem has been fixed. And also that I can continue to plug in more than one appliance in my kitchen!