Was the burnt cable at a junction / onto a component ? From the rest of the post it sounds like a trip used rather than a fuse. I have had major problems with trips in the past and only use fuses now.
If the cable has burnt mid-cable without the entire cable being burnt, then it is most likely where is passes through a thermally insulating area e.g between carpet and underlay, or has passed over an edge which has pierced the insulation and caused arcing.
Volt drop along cable is normal, as all cable has resistance. The thicker the cable, the less resistance therefore less volt drop and less heating, therefore higher current rating.
I'm not familliar with the Mr. Barry kit so without more detailed answers to the above I can't really comment further.