Kitchen circuit died and the connector inside the box looks cooked - how bad is this?

I've fixed two of these melted splice jobs in friends' houses and both times the answer was the same: the connection had been loose for years, slowly arcing under that wire nut until it cooked itself. A wire nut doesn't melt on its own. Bad connection, resistance, heat, and it just keeps getting worse every time you run the toaster or the microwave. So the blob you're looking at is the symptom, not the disease. To answer your actual question, yes, heat damage absolutely travels. The insulation on the conductors gets brittle and cracks well past where you can see charring, and the copper itself can get annealed and brittle. The rule I've always used is cut back until you see bright, shiny copper with soft, pliable insulation, and if that's more than a few inches into the sheathing you're into splice-and-junction territory that gets sketchy fast. Heat like that also wicks down the conductor, so what looks fine can have damaged insulation a foot inside the cable. The unclamped cables are a seprate issue. Cables are supposed to be secured where they enter a metal box, both to hold the box steady and to keep the sheathing from getting abraded on the sharp knockout edges. It's prolly not what caused your melt, but it's a sign whoever wired it was cutting corners, wich makes me trust the rest of that box less. My honest take given what you described: call an electrician for this one. Half a kitchen dead plus a melted splice suggests either an overloaded circuit, a failing connection somewhere else on the same run, or damage hiding in the wall. Outlets and switches are one thing, but tracing a burnt circuit and finding where the real problem lives is a diferent skill level. If you want to do something useful before they show up, leave that breaker off, and take a photo of the melted connector and the box setup so they can see exactly what they're walking into.

This sounds alot like what happened at my brother's place, except it was a bathroom instead of a kitchen. He did the cut back and re-splice thing himself and it worked for a few months, then the same section went dead again. Turned out the actual bad connection was two rooms away in another box on the same run, and the melt he fixed was just where the heat showed up first. So I'd second the electrician route. Quick question for the OP though, does anything on the working half of the kitchen flicker when you run the microwave? Might help narrow down where the problem actually lives.

Been doing this since before wire nuts were even common, we used to twist and tape everything back then, and I'll tell you the other thing to check. Pull that counter receptacle and look at how the wires land on it. The push-in holes on the back of outlets cooked more connections than anything else I ever saw. If the wires just poke into the back, move them to the side screws while you're in there. Also whatever you end up doing, that splice stays in a box with a cover plate, never taped up and buried behind the wall. I've opened up too many ceilings over the years to trust what I can't see.

One thing worth checking before you cut anything: half a kitchen going dead with no tripped breaker sometimes means the counter circuit is a shared neutral setup. Two hots on one 3-wire cable, neutral split between them. I opened up a panel on my buddy's kitchen rewire last year and found exactly that, one hot leg failed and only half the counter worked. If yours is set up that way, flipping one breaker does not make that box dead, the other leg can backfeed through the shared neutral. Get a meter on it, hots to ground and hot to hot, before your hands go in.