Clamp for rope to make in a hoop

Three inch gaps seem quite large - even for a 300 year old wall.

You just need to use a good mortar mix (sand and cement) on the narrower gaps.

Clean out all the old, lose material first to make sure that you get good adhesion and then push the mortar right into the gaps as far as it will go. Work on a small area (1 m²) at a time.

Adding pea gravel to give the mix more "body" might work on the wider ones.

However, I'd get a professional to have a look at the wall first to make sure you're not going to have to redo this work in a couple of years time. If the foundations have gone it might need something more substantial work doing.

Is the wall part of a house or just a garden wall? If it's the latter then dismantling and rebuilding might be the best long term option.

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.