Conceptual intuition in special relativity I am currently studying special relativity at high school and they do a very poor job of teaching it as they just tell us to apply the formulas and give us no basic intuition. Thus I have many basic, conceptual questions that my teachers can't answer:
1) Light clocks seem to be quite a strange concept, why cant we think about special relativity without a light clock? My current feeling on this is they are purely an illustration which makes instantly clear the properties of SR, is this the correct way to be thinking about them?
2) In the derivation of time dilation with pythagorus' theorem and a light clock (the one where you get (ct)^2 = (cT)^2 + (vt)^2), why must it be time that changes? Couldn't the height of the light clock (and thus the distance for the light to travel in one oscillation inside the light clock) change as an alternative way of keeping the speed of light constant?
2.5) The basic idea I am trying to express in 2) (the question above) is: seeing as time is linked to speed through distance, why can't it be distance that changes rather than time in order to keep speed constant?
3) Consider the classic example for an observer standing on a train platform and the train going past at constant velocity. The usual conclusion is that time passes slower for the person on the train. However, doesn't the person in the train see the platform moving with the same velocity and thus conclude that time passes slower for the person on the platform by the exact same amount as previously concluded (but this time in the opposite frame of reference)? In summary, because there is no preferred frame of reference won't the effects experienced by one observer ALWAYS be experience by the other observer and thus all relativistic effects "cancel out"? (Note: this may just be the twin paradox, I'm not sure)
 A: It is easier to answer your second question first.Let's suppose the height of the clock is measured in the y axis and the width in the x axis in some reference frame $O$. The reason the height of the clock does not change is because we assume our boost does not have a component in the $y'$ axis  in the reference frame $O'$ because we stipulate we can only move along the $x'$ axis so there can't be any length contractions measured in the $y'$ axis 
For your first question, why do we always talk about clocks. The simple and flippant answer is because we are just following Einstein and that is how he came up with the theory. The more substantive answer is the following: if I have two reference frames $O$ with coordinates $(x, y, z, t)$ and $O'$  with coordinates $(x', y', z', t')$ and I ask my self the following question, "what is the most general form of a transformation between  $O$ and $O'$ that is consistent with Galileo transformations?" It turns out that it must be able to change the coordinate $t$  in $O$ i.e I can't assume that $t = t'$ .But this means I need two clocks, one for reference frame $O$ and the other for reference frame $O'$. 
A: These are great questions. For your first question, the conceptual answer is that Einstein realized that in every inertial reference frame, we want the laws of physics to be the same. The speed of light is one such quantity which comes about because of fundamental laws of physics (or they were fundamental at the time), namely, Maxwell's equations. For this reason, saying the speed of light must be constant in all frames is a very useful example of a "law of physics" that should stay constant no matter what frame you're in.
Ok, so then why do we construct these weird light clocks which somewhat arbitrarily move transverse to their vertical axis? As another answer mentions: this is because we know the transverse axes cannot shrink or stretch at all (so the $x$ and $z$ axes if the light clock moves in the $y$ direction), but we don't know for sure that the other spacial axis won't change length. That is, we can't really think of a thought experiment that shows the $y$ length should stay constant. Why do we know the $x$ or $z$ axes shouldn't change scale? Well, imagine we have a ring that can fit around a cylinder when stationary. If we move the ring very quickly in the reference frame with the cylinder stationary, then what if the ring's height increases in one dimension? No problem: it still fits around the cylinder and they will not collide. But if we shift reference frames to where the ring is stationary, now the cylinder's height increases and so they should collide in this frame! How can we possibly have two different interactions depending on what perspective we watch it from? Therefore, we cannot have any transverse length changes at all. Like I mentioned before, though, we can't think of any thought experiment like this which shows that length can't change scale along the axis of motion.
This is why we have to consider those weird light clocks: because the way they count time only depends on the passage of time in the frame of the light clock and the height of the light clock! But we know that the height of the light clock cannot change, so these weird light clocks are a very clever way of keeping track of the passage of time in different frames of reference. Einstein was a clever guy, it seems.
For your third question, both will see relativistic effects in each other's reference frame, but they won't cancel out like you are imagining. Usually, the time dilation in one frame becomes "length contraction" in the other frame. So for example, when muons are coming in from outer space they are moving at near the speed of light. The lifetime of a muon at rest is maybe $\tau = 2.2\mu s$, but when the are moving at near the speed of light they make it much further through our atmosphere before decaying. This is because we would watch a clock strapped to this muon to tick way slower, but after it ticks $\tau = 2.2\mu s$ the muon will decay (Ok this is not how decay works but please forgive the inaccuracy for the sake of pedagogy). In the muon's frame, time passes quite normally and it decays after $\tau = 2.2\mu s$, however the "length" of the atmosphere is much shorter in its frame and so it makes it farther down into the atmosphere because it doesn't have to travel as far.
