Electrical Wiring – Electrical Component Melting?

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.

If it smells like burning lint, it's probably burning lint. If the seals are worn around the lint trap (as evidenced by the amount collected underneath), the interior of the dryer can accumulate lint really fast. I'd remove the case and vacuum and or blow out the lint that has collected there.

This would also be a perfect time to give your dryer vent a good cleaning. If it's clogged or severely constricted, this can also cause lint to back up into the dryer chassis.

Either of these situations can be a pretty serious fire hazard, and given that you're already smelling burning, I wouldn't use the dryer until you've done both of these.