My house has a lot of older armored cable using the armor for ground.
If I'm working in a box and find a three-conductor wire with only black/white in use, and I know where the other end is, should I use the red conductor to improve the ground path of that run and its two boxes (also marking both ends with green tape) or should I leave it alone with the red unused and the armor for ground?
That would be a code violation. You may not re-mark green, green-yellow, or bare to any other color, and you may not re-mark non-ground colors to ground colors.
That sounds like old BX cable. It is virtually worthless stuff. I have a number of pieces of it that I occasionally use when rewiring saws and drill presses and other old tools because it looks kind of cool to have multiple armored cables going between switches and motors and suchlike in older tools. I have also used it as feeder wire to fluorescent fixtures rewired with LED bulbs since those no longer have active components and thus are permitted to go 2 wire. Otherwise when you can remove and replace it, do so. Note that this isn't modern armored cable where the conductors are loose in the jacket and can be pulled out and replaced.
If you really want to increase safety in this kind of a system, pull out and replace the panel with a modern panel that is fitted with all GFCI/AFCI combo breakers. The entire point of grounding in home systems was so that the chassis of electrical gear could be grounded so that if something happened inside of the gear that caused the hot conductor to touch the chassis, that the operator would not get a chassis-to-Earth shock through their body. Grounding does not help with people who grab a hot in one hand and the cold or ground in the other.
However modern microprocessor controlled breakers are superior to simply running a ground wire in terms of safety, in my opinion. Of course, a ground PLUS the modern breaker is best but if you don't have a reliable ground, the smart breaker can save your life.
