I've read that this is a common problem when the 74LS125 goes, and mine was so fried it blew a hole in the case such that you could see the die. I replaced it with a brand-new part, and the behavior has not changed.
C4 is not visually damaged and is not a dead short. None of the legs on Q2 are shorted. The drive reads okay until it tries to read a track it has already passed over (and thus erased).
All the pins on the head connector are a dead DC short; is this normal?
Q2 is a transistor, so test it like two diodes, out of circuit. This controls whether you are reading or writing.
The most common thing that causes this is of course the cable being connected wrong, sending 12v in the 5v spot. If left hooked up long enough the damage goes "past" the 74125 and other ICs could have died.