I can see both sides of this topic and since it was never discussed on screen we will never know. But I don't ever recall seeing a TOS PT tracking or maneuvering or in anyway being guided to a target.
But if the torpedo weren't tracking and maneuvering, it would never hit anything, except perhaps stationary, gigantic targets at point blank range.
We don't know the method of tracking - is there always a sensor at the tip of the torpedo, or never a sensor there but only at the starship? But guiding is clearly required, and that can't happen without tracking.
Or without maneuvering. We don't know the method of maneuvering, either - but we do see the glow, which quite possibly is simply the exhaust from the rocketlike holes at the stern. Might be a rocket flame giving Newtonian thrust (like in today's missiles). Might be mere venting of wastes so that the internal engine can produce its non-Newtonian thrust using invisible fields (much like in steam-propelled torpedoes of yore vented the steam nonpropulsively while the propeller pushed the torp forward). Might be the propulsive field itself (although it's basically never visible in any other spacecraft of Trek).
As for observed performance, the visuals only show the launch. We can't tell if the torp typically executes a 782 degree turn after launch just for shits and giggles, but we can tell in "The Changeling" that Kirk fires over his shoulder (the bearing to the target is given as being behind the ship (and not directly behind) while the torpedo initially flies forward).
I also wonder also if they were shown (in TOS) as being used at all speeds or just at warp?
Yuppety-yup, they covered those bases in the original show already. The ship was in orbit in "Arena", prevaricating over whether to beam up Kirk and pals, when torps were first mentioned used in Trek. Warp firings in turn were evident e.g. in "Elaan of Troyius".
What they didn't show was using torps for ground bombardment. That came later, in TNG and the TOS movies.
Of course, the visuals always showed the torps leaving the ship at a snail's pace, certainly nowhere near lightspeed. So either we saw the combat scenes in slow motion; the torps accelerate after launch on their own; or we saw the combat scenes in fast-forward motion, omitting the minutes (or sometimes weeks!) it took for the torps to find their targets.
Timo Saloniemi