Yes on the 'when the doors are opened' comment.
The glove box light is independent of the courtesy/under dash lights. There may be some wire harness junctions that are disconnected.
You need a copy of the 65's wiring schematic to trouble shoot the system. Although not part of the 65 Shop Manuals I've seen, it is available separately from vendors.
Charlie, CMSGT USAF Ret. 65 Hi-Po F/B, 67 GTA F/B (SB) Mustang Owners Club of Austin (MOCA) TX, MCA 27479