Unburdened dwarves got the short end of the stick here. Every other race gets a bonus that they didn't have before from these new heritages. Unburdened dwarves get to spend their 'free' heritage to get back what was taken away in the same update. Were dwarves too overpowered before that all other races needed a boost to get them back in line?
If you are stabilized by either the Administer First Aid or the Stabilize spell, you lose the dying condition, but remain unconscious at 0hp.
If you gain hp through healing magic, a potion, a Battle Medic or some other way, you lose the dying condition and gain consciousness, but are slowed.
Yes, if someone is unconscious, at 0 hp, not dying, and receives hp healing they gain consciousness and are not slowed.
The first Quote of your misses the sentence right before it:
You lose the dying condition upon returning to 1 or more Hit Points. When you lose the dying condition, you regain consciousness, but are slowed for 1 round.
The text in Dying is the general rule, the rules in "Administer First Aid" and "Stabilize" are specific rules that are the exceptions.
If losing the dying condition by itself is not sufficient to become conscious, those two sentences need to be combined into one to make that clear. Something like "Upon returning to 1 or more Hit Points, you lose the dying condition and regain consciousness, but are slowed for 1 round." The way it's phrased now makes it sound like losing the dying condition is the reason for regaining consciousness, but it's not. It's regaining the Hit Points.
On page 3 of the Pathfinder Playtest Rulebook Update 1.2, under Dying it says "When you lose the dying condition, you regain consciousness, but are slowed for 1 round."
Under Administer First Aid on page 4 it says "The significant changes are that the creature remains at 0 HP (now that you can be at 0 HP and not be dying)" and then under the skill Administer First Aid it says "Success The creature at 0 Hit Points loses the dying condition (but remains unconscious), or you end the persistent bleed damage (see page 323)."
Under Other Changes on page 4, it also says "In the stabilize spell, change the description to “The soft glow of positive energy shuts death’s door. The target loses the dying condition, though it remains unconscious at 0 Hit Points.”
Since these are the two main ways of losing the dying condition other than receiving healing, do you regain consciousness or not when you lose the dying condition?
Another question I have is that when you regain consciousness, you are slowed x for one round, where x is the dying value you had when you regained consciousness. If you don't regain consciousness from Administer First Aid or Stabilize, what if I apply either of those to a dying character to remove the dying condition, and then with my next action heal them. Since they lost the dying condition before they regained conscousness, are they not slowed for one round?
I'm not sure if making your saving throw wakes you up. There's nothing to indicate that it does, it just says that your dying condition goes down. Just about every other instance of 'stabilizing' leaves you at 0 HP and unconscious, so I'm not sure why it wouldn't.
It is poorly-worded, but: "When you lose the dying condition, you regain consciousness, but are slowed for 1 round."
And: "When you’re unconscious and at 0 HP but no longer dying, you naturally return to 1 HP."
I believe you are supposed to return to 1 Hit Point.
I think the sentence "When you lose the dying condition, you regain consciousness, but are slowed for 1 round." is supposed to be additional information about the sentence before it: "You lose the dying condition upon returning to 1 or more Hit Points." It should read, I believe, "Upon returning to 1 or more Hit Points, you lose the dying condition and regain consciousness, but are slowed for 1 round."
The way it's written doesn't imply that, but under Administer First Aid, it says "now that you can be at 0 HP and not be dying" and the mechanics text of Administer First Aid specifically says "loses the dying condition (but remains unconscious)"
As for your second quote, you left off the second half of the sentence: "When you are at 0 HP but no longer dying, you naturally return to 1 HP and awaken after sufficient time passes" This also implies that losing the dying condition doesn't immediately put you at 1 HP and conscious.