For official rules I'm pretty sure Caliber has it right, this is cut and paste directly from the Quick Start PDF (I know, not necessarily the best rescource
) I haven't seen anything about auto-stablizing at the end of combat, though I am no 4th edition rules Guru. I'm interested to see what Legildur may find.
Healing in Combat
Even in a heated battle, you can heal. You can heal yourself
by using your second wind, an ally can use the Heal
skill on you, and an ally can use a healing power on you.
When a power heals you, you don’t have to take an
action to spend a healing surge. Even if you’re unconscious,
the power uses your healing surge and restores hit
points. And some healing powers restore hit points without
requiring you to spend a healing surge.
When you are dying, any healing restores you to at least
1 hit point. If someone has stabilized you using the Heal
skill but you receive no healing, you return to maximum
hit points after an extended rest.
(Emphasis mine)

Healing in Combat
Even in a heated battle, you can heal. You can heal yourself
by using your second wind, an ally can use the Heal
skill on you, and an ally can use a healing power on you.
When a power heals you, you don’t have to take an
action to spend a healing surge. Even if you’re unconscious,
the power uses your healing surge and restores hit
points. And some healing powers restore hit points without
requiring you to spend a healing surge.
When you are dying, any healing restores you to at least
1 hit point. If someone has stabilized you using the Heal
skill but you receive no healing, you return to maximum
hit points after an extended rest.
(Emphasis mine)