Yeah, a faulty ground will cause all kindsa goofy problems. Gauges will fluctuate, lights come on and go off, it's like a thing possesed.
That little braided wire isn't really a ground, it serves as a sort of inductance shield. I'm not sure about thexact purpose, but I'm sure it has something to do with the ignition system, and shielding r/f signals emitted by it. If it's getting hot, then it's trying to act as the jumper to ground, and it's not heavy enough for that.
Replace the ground to body, and you should be OK. The battery drain could be any number of things, the alternator may be a good suspect, but I'd replace the ground first.
Good luck!
-Taiji