Why does time dilation even occur? Confusion I was trying to understand time dilation, but ended up getting more confused.
Why does it occur.
I saw the Wikipedia visualization here, which explained that a greater distance led to greater time (since 2nd postulate states that velocity is constant)
After seeing this, I had a list of questions:
  1. What is so special about the speed of light to make time dilation depend upon it?
  2. Why does time dilation even occur?

For question 2, I thought of a counter-case, which I have described below:

Consider a projectile horizontally launched at some vertical height. Now, consider another identical projectile dropped from the same height. Simple Kinematics tells us that the time of the projectiles in the air will be the same.

In this scenario, the projectile horizontally launched traveled a greater distance when compared the ball dropped (similar to the Wikipedia image). The time it took was identical. But the analogous time dilation example shown by Wikipedia suggests otherwise.
I would really appreciate it if you could help me sort out this dilemna with question 2 especially (tell me where my thinking is off), and also explain the answer to question 1.
 A: Time dilation is best thought of an apparent effect, an illusion caused by mixing a historical or classical view of time with a fully relativistic one. Historically people have thought of time as a global concept, formalised by Newton with his notions of Absolute Space and Absolute Time. But, strictly, this is an assumption useful in Newtonian mechanics which cannot be fully justified empirically. What does time on Earth mean for time on Andromeda? How can you say what time is now on Andromeda? Our experience of time is personal to ourselves, and, similarly, when we read the time from a clock we can only say that this is the time local to the clock. Time local to the clock is the proper time of the clock.
The fundamental principle of general relativity is that the laws of physics should be the same in all reference frames. Then all identical clocks measure the same unit of proper time, and run at the same speed, one second per second, no matter where they are or how they are moving. Likewise constancy of the speed of light follows from the general principle because it happens that light moves at the the fastest possible physical speed.
The issue comes when we compare the proper time of different clocks, in different places or moving at different speeds. Then we have an apparent effect of time dilation because we are not comparing like with like. In special relativity, the definition of synchroneity depends on how clocks are moving, and in general relativity it is affected by geometry.
The geometrical effect can be compared to two ships moving at the same speed due East, one on the equator, the other at latitude 060. If their paths are plotted on a Mercator projection, it will appear that one is moving twice as fast as the other, although we do know that this is not actually true.
A: One of the postulates of special relativity is that the speed of light is the same in every intertial reference frame.  For this to be true, either lengths have to contract or time has to slow down, or both.
If you assume that the speed of light is affected by the speed of the inertial system, then this will conflict with the postulate.
From Symon, Mechanics, Third Edition:  We see that in order to resolve the paradox [between the postulate and what seems to be common sense in adding velocities] we must suppose that when one coordinate system moves relative to another, either the units of time or length, or both, are different in the two coordinate systems.
To my mind, it does not make sense to ask why either of these occur.  They do, and they are a consequence of the postulate concerning the speed of light.
Edit: I just looked at the Wikipedia link.  What is shown there is a common demonstration of time dilation.  Symon goes through this example in detail.  The example only demonstrates something if you think about what is seen from two different inertial coordinate systems.  (Sorry, it is too involved to go through in a post.)  But in your example, you only are thinking about what happens in one.
Special relativity is largely concerned with understanding the proper way to change coordinates between to inertial frames of reference that are in motion relative to one another.
A: 
Consider a projectile horizontally launched at some vertical height. Now, consider another identical projectile dropped from the same height. Simple Kinematics tells us that the time of the projectiles in the air will be the same.

The horizontal motion slows down as the vertical speed increases.
If the projectile is a pulse of light, then always:
$$  \sqrt{v_x^2 + v_y^2} = c   $$
If $v_y$ increases, then $v_x$ must decrease
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Just a very small change is needed to get increased falling time: Launch the planet under the projectile too.
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As we observe the launched projectile slowly increasing its speed towards the launched planet, we may ask: "why does the speed transverse to the falling motion not change at all? Like, shouldn't time dilation decrease the transverse speed?".
Well, I guess Simple Relativistic Mechanics does not answer that "why not" question. It just simply tells us how objects move.
