Electrical – Why doesn’t the main breaker shut off power for house

I'm not in the US, so I used Google to deduce which prong is doing what in this outlet. I'll assume that the left and the right wires are phase wires and the middle wire is the ground.

Note that although the left wire looks melted the right one looks just fine although they usually carry equal currents. So this is not melting due to overload. The most likely scenario is that the left terminal got rusty and its surface resistance raised and so it overheated and this causes minor melting on the left wire.

The signs of burn on the outlet face are quite typical and likely don't indicate a wiring problem. Outlet contacts tend to get loose over time and once that happens they no longer maintain tight contact with the plug contacts. Also when a high load is disconnected an arc can start and this makes the contacts burn and cover with carbon and further increases the resistance. This is more likely to lower quality outlets, but happens just about everywhere.

So the bottom line is the outlet has just worn out and needs careful replacement. Remove the old outlet, clean the exposed wires, carefully connect the new outlet making sure all screws on the terminals are tight. You might want to test the new outlet - plug some high load and let it run for an hour, then trip the breaker, open the outlet cover and check whether there're any signs of warming up.

Your box, sadly, is toast -- I suspect the mismatched circuit breakers (you have an ITE/Siemens box which uses type QP breakers, yet there are BR and HOM aliens roaming around in it, which is bad as while 1" breakers may appear to fit in other folks 1" type panelboards, they certainly won't fit well!) are part of why you're seeing the arcing and burning on the bus tabs.

As to what's going on with breakers 9 and 11? Have your electrician measure for a short between the two circuits with them both disconnected from their breakers in the panel and tagged. (If he's really good, you might be able to get them meggered even -- this will find "leakage paths" between them in addition to "hard" connections, but requires gizmos to be unplugged and light bulbs removed from the circuits lest something fry from the HV an insulation tester applies to the circuit under test.) If you see a short or leakage path, then that's your culprit, and it's time to basically trace, divide, and conquer at that point.