Oh man, sounds like a nightmare. Poor little Valiant. If he shorted the car badly enough to actually burn the cable up ad melt the terminal off the battery then it was a serious problem. First off, I would replace the starter relay. Who knows what has meltd together in there.
Second, I would pull apart the bulkhead connector to see how many of the terminals have melted together. The Mopar bulkheads have a bad habit of ovrheating and corroding anyway, and with a ajor meltdown like what happened in that Valiant who knows what wires got crossed.
If I remember correctly, the engine should die if you pull the positive battery cable off the battery since the charging circuit is broken. The fact that the engine still runs off the alternator indicates to me that there is a short between wires somewher ein the charging/starting system. This also relates to the grinding noise.
I suspect the grinding noise is the alternator freaking out under some kind of shorted load. The bearings on the alternator are probably worn to the point that the unusual load put on the alterntor makes it chatter, or perhaps some of the field windings in the alternator have been shorted or reversed causing thre to be reversed polarity in the fields when the battery cable is removed. I used to know a lady who oned a 63 Dart with a slant that I maintained for her. I converted her over to electronic ignition and a dual field alternator. Several month after I did the switch the car started making this horrible grinding noise while running. She took it to a mechanic who told her that the engine needed to be rebuilt. In a panic she called me and I took a lok at it and discovered that it was just the bearings on the altrnator which had gone out. I swapped out the alternator with a spare I had in the garage and it was as good as new.
If the transmission sifted fine ad you don't see any obviously problems laying in the pan then I would think it is okay. Torqueflites generally start to slip really bad before they make any grinding noises.
All in all, I don't envy your situation. I see two easy fixes: (1) new/replacement wiring harness, or (2) if you are up to the project, create your own wiring harnes but upgrade it using better connectors and relays.
The upside is that if you have a points style ignition, you can upgrade to 70s electronic ignition and the dual field alternator by carefully removing the alternator, VR, distributor, ballast resistor, and all associate wiring back to the bulkhead connector. Those two systems all come down to the brown and dark blue wires passing through the firewall. If you ar careful you can just remove the terminals form the bulkhead connector of the donor car and plug them into the one on the 67 and just re-wrap the wiring harness.
Any way you cut it you have some major wiring adventures ahead of you. I will dig through my shop manuals and try and find a 67 wiring diagram for you. I should have one in my Chiltons.
You mean a maybe it's the correct (typical) diagram?