I’ve shot myself in the foot by responding to relativity questions before, but it’s an interesting topic, and I love mind puzzles like this one introduces.
There’s no absolute frame of reference for time any more than there is for space. Any measurements you take have to be relative to something. If you’re in a lab on one spaceship viewing a lab on another spaceship, your frame of reference for direct measurements is where, and when, you are.
Its perhaps counterintuitive that you, in your lab, see your brother Bob’s clock running slowly, and Bob, viewing you, sees your clock running slowly, even though as far as you can measure locally your clock is running just fine. But it’s true.
It should at least be reasonably obvious that if your question #1 is true, that all motion is relative, that the situations should be symmetric. If you are moving away from Bob and see his clock moving more slowly, then Bob is moving away from you, and he should see your clock moving more slowly. This defies common sense in that it means each person is running at ‘normal speed’ while the other clock is slowed down.
Relativity is chock full of surprises like this. What it shows is that time, like space and motion, are local concepts. It also shows that there’s no sense of a universal clock. That means there’s no tick-tick-tick out there by which you can universally measure when something happens.
The problem with assuming that you don’t see Bob’s clock slow down because you’re also slowed down is that it assumes there’s a universal clock by which you can measure that both you, and Bob, are slowed down. It assumes that you are slowed down by exactly the same amount as Bob no matter which frame of reference you’re in. But the moment you start making assumptions about absolute frames of reference, you start running into trouble with the speed of light being constant, which we’ve demonstrated in many experiments to rather stupifying amounts of accuracy.
Actually, time dilation itself has been directly measured, so the question has to become: Well, this suggests I can detect Bob’s clock slowdown, so what’s wrong with my assumptions?
Here’s a thought experiment which partially shows that time can seem to slow down for someone other than yourself. For those relativity/math gurus, note that I’m ignoring Lorentz contraction in the direction of motion for the moment.
Assume that the speed of light is constant.
Imagine that you’re in a spaceship and you pass by Bob, in another identical spaceship at a point p0. You’ve arranged to pass by him at 50% of the speed of light. You’ll pass by him at a time we’ll call t0.
Let’s assume you both have a clock on board which, from the point of view of the person on that spaceship ticks away every second. At every tick of Bob’s clock, Bob sends out a light pulse to you (and you to him each one of your pulses, but ignore that for the moment).
The clock indicates a passage of time. In the normal course of operations, you can measure everything Bob is doing according to the passage of time of his clock, as much as you can measure everything you’re doing in terms of the passage of time of your clock. If it takes you ten minutes to play a game of solitaire by your clock, and Bob’s your (twin, identical) brother, he can play the same game of solitaire in the same time. That is, your local clock can be considered a measurement of the continuous passage of time even though the ticks are discretely 1 second apart.
So, you’re watching light pulses from Bob as you pass by him. Those light pulses correspond to ticks of Bob’s clock. That means the light pulses are indicative of the continuous passage of time on his ship. That game of solitaire you play that takes 600 ticks of your clock (10 minutes) should take Bob 600 ticks of his clock, or 600 pulses of light from his ship.
But you’re rocketing away from him. Even if your time frame is personally slowing down locally on your spaceship you’d be unaware of it because you’re slowing down too, so your game of solitaire is still 600 ticks of your clock long. So local time on your spaceship is still ticking away merrily at the normal rate.
So the only thing that matters is relative time – that is, how are you perceiving Bob’s passage of time? Your own time must by definition seem normal, so the only question of interest is what’s happening aboard Bob’s ship as viewed from yours.
You know your ship and Bob’s are identical, as are the clocks. So you know his clock should be ticking away at what Bob considers to be the normal rate.
So let’s assume for the moment that Bob’s rate of time passage is the same as yours.
This means that he’s sending out pulses every second. At your time t1, one second after passing him, he sends out a pulse. This pulse races away from him at a fixed speed, the speed of light. The pulse is heading towards you. You’re at P1 at the time this happens.
At your time t1, though, you’ve moved away from him by a distance of 0.5C1 (pick your units… 0.5C meters per second, times 1 second, equals 0.5C1 meters). So the light’s going to have to travel this distance to get to you, and since the speed of light is constant, this will take a known amount of your time. In fact, it will take 0.5C*1/C, or 0.5 seconds.
But in 0.5 seconds, you’ll have moved away from P1. You’ll be slightly further, so the light will have to travel even further. This additional distance will take 0.25 seconds. But in that time you’ll have moved even further… and so on. The gap is getting smaller, of course, and the light pulse from Bob will eventually reach you, but it won’t reach you at a distance which is 1 of your seconds of travel at 0.5C. Wherever this point is (you could work it out, but it doesn’t really matter where it is), it is after P1. Call this point P1’.
P1’ is strictly further along in space than P1. Since P1 is the distance covered by your spaceship at half the speed of light, P1+dP represents P1’. dP represents this extra distance. But the speed of light is constant, and your velocity is well known relative to Bob. (P1+dP)/C gives the time it takes, therefore, for the pulse to reach you.
But C is constant, and the distance you must have covered before the pulse got to you is greater than the distance you should have covered in 1 of Bob’s pulses if your clocks are passing time at the same rate. Bob’s pulse is covering longer distance because you’re moving away from Bob. But because the speed of light is constant, this implies it takes a longer time for the pulse to reach you.
This implies either that your velocity isn’t what you thought it was, or that Bob’s pulses aren’t coming out every second.
However, your velocity was established as part of the thought experiment, and you never underwent acceleration.
Therefore, Bob’s clock isn’t ticking every second, from what you can tell. It’s ticking slower because the light is taking extra time to reach you, so the events of the pulse reaching you are separated by more than a second.
You can argue the same thing about the arrival of the second, and all subsequent pulses.
Bob’s pulses, however, are indicative of Bob’s clock, and the ticks of Bob’s clock describe his continuous passage of time. It’s not just that the pulses seem to arrive slower… this implies that the clock itself is ticking slower, and so everything that Bob does that you could possibly witness from your vantage point must be moving slower.
You can also argue about all of this about Bob’s interpretation of your time relative to his (its symmetric). This means that Bob must perceive your clock as running slowly.
This just goes to show that the apparent passage of time is a local phenomenon, and that there is no universal clock. Each of you is no more right than the other. If you’re both passing time normally (to yourselves), but the other appears to be passing time slowly, then there can be no universal measure by which to analyse events anywhere in the universe.
Sadly, this isn’t quite the way the universe works. The math above is slightly bogus. If you analyse the light pulses in this way, you’ll come to the incorrect conclusion that if you’re both heading towards each other that you’ll see each other running faster. This isn’t the case. You’ll still see each other running slower.
The problem is that in the direction of motion, as you approach the speed of light, there’s an effect called Lorentz contraction that monkeys with relative distances – but only in the direction of motion. This means that some of the time dilation effect described above is accounted for by Lorentz contraction, and it is this effect that changes the nature of the approaching-your-brother case.
So all I meant to describe above is to show that you can have a situation that describes the fact that time dilation can exist. The math above is not meant to show how much dilation occurs.
You can, however, set up experiments (like the light-clock-on-a-train thought experiment) that shows the true time dilation effect by constructing light-motion measurement that is perpendicular to the direction of travel and therefore is not subject to Lorentz contraction effects.
There, I’ve put my understanding under the microscope of the dopers, who will most undoubtedly shoot many holes in this. 
