Both dnd-3.5e and pathfinder lack a formal definition of what a turn is. However, its meaning could be inferred from other rules.
If we all agree that in a round, every combatant takes a turn and in your turn you take actions, then I'd infer a possible answer using the following hints (bold and italic emphasis is mine):
Free actions don't take any time at all, though there may be limits to the number of free actions you can perform in a turn.
In general, speaking is a free action that you can perform even when it isn't your turn. Speaking more than a few sentences is generally beyond the limit of a free action.
[...] You can perform one swift action per turn without affecting your ability to perform other actions. In that regard, a swift action is like a free action. You can, however, perform only one single swift action per turn, regardless of what other actions you take. You can take a swift action anytime you would normally be allowed to take a free action. [...]
The latter sentence strengthens in me the idea that free actions (like swift ones) can be usually be taken in one's own turn. I know that this is not a strong implication (the sentence really vehicles the concept in the other way around), but it gives me a hint.
[...] However, unlike a swift action, an immediate action can be performed at any time—even if it's not your turn. [...]
So a special definition has been forged for actions that can be taken outside your own turn. Is it necessary in order to introduce the hard limit of 1 per round? Or does it sit there also because not even free actions can be taken on other turns (besides speaking a few words)?
Now I opt for the latter, then:
Scenario 1: No, unless rule-0 overridden or by exhuming the Instant Rage feat, a Barbarian cannot enter rage outside of her turn. Likewise, a character cannot actively drop prone outside of her turn.
Scenario 2: When the attack of opportunity is resolved, we are in the Barbarian's turn. Leveraging the definition of free actions (free actions don't take any time at all) I'd say that the Barbarian can enter rage in response to the damage taken by the attack of opportunity and still keep swinging her sword. This is enough, in my opinion, to supersede the dying condition immediate effect (character immediately falls unconscious and can take no actions).