To answer your last question first, there are no fuses in a normal plug socket.
There are some special fused sockets that DO have fuses but they look different to an ordinary 13amp socket. They will have a rectangular cover in the socket faceplate which holds the fuse inside the socket and as a notch at one end. You get at the fuse by levering it out (carefully!) with a screwdriver in that notch.
Often the fuse cover-cum-holder will be in red rather than the white or cream of the rest of the face plate, or it will have ‘FUSE’ written on it. And with modern wiring systems they won’t be sockets that you can put a three-pin plug into either, they will have the cable for the appliance wired directly into the socket.
These fused sockets are used for connecting permanent fixed appliances, often in the kitchen such as cooker hoods.
Problems With Sockets
However, it doesn’t sound like that’s what you’ve got so on to your other questions. Inside a plug socket there are brass connections that wires are fixed to, much like there is in an ordinary 13amp plug. These take the current to and from the socket and connect to the plates that the pins of the plug slide in to. Otherwise the only other active part of a socket is a connection for the earth, which should have the single copper core of the electrical cable screwed to it.
There is one socket still working, a 30amp one, but 30amp sockets should be on a separate circuit back to the consumer unit (fuse box) from all the 13amp sockets. So although it looks like this 30amp socket is on the same circuit it almost certainly isn’t.
Don’t Mess With Sockets unless You’re an Electrician
That would mean that there is something wrong in one of the sockets or in the wiring somewhere on that ring. You might find that some of the screws are loose and that a short is overloading the fuse of the circuit breaker that serves that particular ring.
However, you should not do this – you need to get a qualified electrician in.