[RPG] Can you heal yourself at 0 HP

The Spectral Hand spell is notoriously unclear. Nevertheless, I'd say you're right regarding this particular point.

1. You mentioned the real selling point yourself, "attacking with the hand counts normally as an attack". This gives it a place in the 3.5 action economy, the one of replacing an attack. Importantly, there is nothing suggesting the spell requires other actions.
2. There is no mention of the Spectral Hand spell ever being directed. It is an effect that hovers near you, the act of using it to deliver touch spells is not referred to as "directing" it.
3. Spells that require (re)directing say they do. Look at Spiritual Weapon or Flaming Sphere.

The Special Edition PHB disambiguates the wording. Some people refer to this as "stealth errata" since it doesn't seem to have made it into any other sources, nor was it conspicuously announced anywhere. Even the SRD doesn't seem to have been updated to contain the new wording. Nevertheless, it makes the intent of the feat clear. Emphasis mine:

When using the attack action with a melee weapon, you can split your move action in that round in order to move both before and after the attack, provided that your total distance moved is not greater than your speed. Moving in this way does not provoke an attack of opportunity from the defender you attack, though it might provoke attacks of opportunity from other creatures, if appropriate. You can't use this feat if you are wearing heavy armor.

You must move at least 5 feet both before and after you make your attack in order to utilize the benefits of Spring Attack.

Notably, this is still not the same as Pathfinder's implementation: instead of combining the attack and move into a single full-round action, it keeps them as two distinct actions but allows you to apply them in an unusual way (much like the Overrun writeup that you cite). But neither implementation allows you to make a spring attack using a single standard action.

If you want to houserule Spring Attack to take a single standard action, go right ahead. No one will stop you. I might consider such a thing myself in my game, if it ever comes up. But that's not RAW, and in one of those rare cases where RAI is demonstrable, it's not RAI.