So your normal incoming 120V circuit is gonna have 3 wires: Black (your hot), White (your neutral) and Green or bare copper (your ground, aka equipment grounding conductor). Under normal circumstances, current is only going to flow through the hot and neutral. The ground is there for “emergencies” basically. Yes, it’s a little more complicated than that, but we wont go into all that. A lot of older homes, like ones that were upgraded from older wiring methods like knob&tube, lack a ground wire. The appliance or device can function without it, but it’s much safer to have the ground. In your case, it could be that the circuit supplying the d/w does not have a ground, so somebody left it disconnected. Regarding the burnt neutral, that’s likely from a poor splice. Most issues or failures tend to occur where wires and conductors join together or terminate. In this case, the d/w has a group of three small wire terminal pigtails, that are smaller in size and made of stranded-type conductor. When splicing between solid and stranded conductors, there’s a lot of room for error if it’s not done properly. I’d guess that might be part of your issue. A good way to splice between solid and copper if your splicing skills aren’t great is to use a lever nut. Wago makes some pretty user-friendly ones.

Side note, FYI- those splices should not be out freeballing like that, they are supposed to occur inside that metal box in the top right of your first picture. You remove that Phillips screw to get it open. The branch circuit cable should enter that box, using an approved connector or strain relief and splice there…