Doing laundry and thinking about a specific barbarian class, I didn't think about the literal interpretation of the rule coming into play. If the judge wants to let a guy jump into lava and swim around, then I suppose that's their business. I don't think that would be the case in the main, however.
Allow me to refocus my comment, as it was not so much about what a judge might actually allow but about how there is a clear opportunity to put a simple limit (other than just time) on how much abuse a barbarian can use To the Death to fight through - some examples: 20 extra hit points; fight through any amount of HP damage but still die from any effect that causes instant death; simple return to full hp and die permanently upon reaching 0 the 2nd time.
I think simply leaving it to a time limit, which again I question why you went with a time limit that differs from the normal time limit for characters that are bleeding to death, is too powerful mechanically.
Narratively, I see no point to the feature - any character can perform the narrative act of sacrificing himself holding off some big bad while everyone else escapes, it's a feature of the system's rules about withdrawing from combat.
I had in mind the third edition fighter progression, character level as base attack bonus. I have nothing against the 1d3 progression of the dwarf and the warrior. I love the deed die! But the barbarian above is hitting less often, but more brutally when the hits do land.
I don't see a need for a 6th attack bonus progression - so if you don't want the barbarian to have deeds, then I'd recommend sticking to the Halfling attack progression.
As for "hit less often but more brutally when the hits do land," you may actually want to use Deeds - the deed die adds to damage as well as to attacks, so the Warrior has better damage potential than your barbarian does as currently written.
Ultimately, I want this class to function similar to the warrior, but without usurping the deed die mechanic that makes the warrior really exemplary. The barbarian doesn't get the flashy moves, the barbarian bashes into the fight, and bashes until the fight is over. I think the above does a good job of illustrating that mindset.
I think the warrior does the same thing, if that's the way you flavor it.
In a way I think that each feature you include is a decent idea taken too far. I only speak from my own experiences, but here are some things I would worry about were I to include your barbarian class at my own table.
Unarmored bonus AC from Stamina - a decent way to encourage wearing no armor, which is all well and good but I am personally avoiding... I'm of the opinion that Fumble Die is the only incentive needed to encourage players to tend toward light armor or no armor, and Conan (the constantly shirtless barbarian) wears armor when he can afford (financially and circumstantially) to do so.
Speed increase - not a bad rule, though I am not a fan of conditional bonuses since that means that some barbarians (those that won't have "good enough" defenses without armor) will never benefit from it.
Song of Battle - too situational in my opinion, a party member trying to "help out" as normal can actually hinder the barbarian by taking his kill and preventing any damage reduction kicking in... I'd recommend finding a flat rate that applies without need to rack up killing blows, especially without any ability enhancing the barbarian's damage output.
Also, the high-end of the power is too powerful... high level combat would be required to only include limited numbers of potent creatures as the barbarian would be invulnerable, or nearly so, to the attacks from your standard "entire army of orcs" type encounters.
I would recommend a progression that increases at the same level as a warrior's critical hit range, and is simply a single point of damage reduction added at each interval (1 at 1st level, 2 at 5th level, 3 at 9th level).
Trust no Weavers of the Dark Arts - the benefit of the feature (gaining temporary hp) doesn't really match the flavor - mistrust of magic and added ability or focus against it... especially because it could come up, but then do absolutely nothing, if the offensive magic comes from the only combatant in the battle - there is no one around to smash for minor temporary hp gains.
I think it should have some effect against magic itself (a bonus to saving throws or a % chance to resist regardless of saving throw results or allowances for example) or against the user of magic directly (a bonus to attack or damage against the user of the magic that triggered the effect).
Final thought: Are you sure you even need a new class called "barbarian" and not just a warrior who is a barbarian by flavor?
You seem to want a big, tough, hard hitting type - that's exactly what warriors already are.