Why does the article say that clocks "appear" to be running differently? Are they running differently? Or they only appear to do so?
If so, what happens when the two participants get closer? Do clocks start "getting in sync"?
If we had an atomic clock with a mechanism to self-destruct when not running at the original "right" time, does the atomic clock self-destruct? Can it "tell" it is running slower?
In more biological settings (say, a human being) what are the effects of "aging faster"? Is it possible that the body would "not work correctly" under certain gravitational forces because of the effects of time? (i.e. maybe the blood flows faster, just a terrible example, I hope you get the idea).
> Are they running differently? Or they only appear to do so?
As acchow says, this question doesn't really make sense, but if you want a short answer, I agree with his: they are running differently.
> what happens when the two participants get closer? Do clocks start "getting in sync"?
I don't understand what you're asking here: can you be more specific about what "getting closer" means and what "getting in sync" means?
> Can it "tell" it is running slower?
If the clock can only use measurements internal to itself, then no, it can't tell. It needs to be able to compare itself to some external reference (such as another clock in a different state of motion) to "know" that it is running slower. And even then, all it "knows" is that it is running slower relative to whatever other clock it is comparing itself to.
> Is it possible that the body would "not work correctly" under certain gravitational forces because of the effects of time?
Not because of the effects of time alone, no; those just make the body age differently relative to a similar body that is in a different state of motion or at a different place in a gravity well. Gravity can cause a body not to "work correctly" in other ways, such as tidal gravity disrupting its structure, but that's not an "effect of time".
> Why does the article say that clocks "appear" to be running differently? Are they running differently? Or they only appear to do so?
Short answer: they are running differently. Complicated answer: I don't think this question really makes sense in relativity.
> If so, what happens when the two participants get closer? Do clocks start "getting in sync"?
No.
> If we had an atomic clock with a mechanism to self-destruct when not running at the original "right" time, does the atomic clock self-destruct? Can it "tell" it is running slower?
There is no "right" time in relativity, only frames of reference.
> In more biological settings (say, a human being) what are the effects of "aging faster"? Is it possible that the body would "not work correctly" under certain gravitational forces because of the effects of time? (i.e. maybe the blood flows faster, just a terrible example, I hope you get the idea).
Relativistic time dilation will have no effect on chemical and biological reactions. But I'm sure there are aspects to our biology that go beyond chemical reactions and actually rely on quantum effects - time dilation might have some effect here? I don't have the answer for this.
> Relativistic time dilation will have no effect on chemical and biological reactions.
This is not correct. Relativistic time dilation affects all processes. If you flew a chemical reaction and a living organism on a GPS satellite, and then brought them back and compared them with a similar reaction and organism that stayed behind on Earth, the reaction on the satellite would have proceeded further and the organism on the satellite would have aged more. (In practice, the differences would be too small to detect with our current technology, but if we had accurate enough ways of measuring we could detect them.)
> That's the actually movement of time changing. The processes themselves are unchanged
You can interpret things this way, I suppose, but that doesn't change the fact that it applies to all processes; from the viewpoint of someone who sees the processes as moving, all processes are time dilated--the "movement of time changing" applies to all of them. There's no special exception for certain quantum effects, or anything else. They're all the same: they all are unchanged to someone moving with them, and they all are time dilated to someone not moving with them.
If so, what happens when the two participants get closer? Do clocks start "getting in sync"?
If we had an atomic clock with a mechanism to self-destruct when not running at the original "right" time, does the atomic clock self-destruct? Can it "tell" it is running slower?
In more biological settings (say, a human being) what are the effects of "aging faster"? Is it possible that the body would "not work correctly" under certain gravitational forces because of the effects of time? (i.e. maybe the blood flows faster, just a terrible example, I hope you get the idea).