When the two waves collide, why do they pass right through each other?
The problem in understanding waves, in my opinion, lies in the fact that one often applies the same concepts he uses in describing particles, to waves. Waves are not particles, and particles are not waves.
While this seems a stupid tautology, it's not that easy to stop mixing the concepts and start thinking in the right framework.
Mathematically it's due to the principle of superposition: the sum of the two solutions of a wave equation is also a solution.
Superposition principle is way more fundamental than you could think. It doesn't just tell you that the sum of two solutions is a solution; it tells you that you can think about each wave independently from the other waves, as if the weren't there. You can picture each wave travelling down the wire, and then sum all the waves that compose your whole waveform.
But intuitively it's not clear why the waves would not, say, just cancel each other during the collision.
Start reasoning in terms of waves. How could a single wave be stopped? Only by dissipation in the medium, or by external forces. Not by means of other waves. That's where the superposition principle plays its role. Different waves (in a linear medium) no not interact. That's it. In your example you see a sort of interaction, but it's actually just a coincidence. It's just visual. You are interpreting the waves as interacting, but actually they are ignoring each other and keeping their behavior unchanged. You could test my statement analyzing the two waves in terms of their momentum/wave-vector instead of their "position".
What would be a convincing 'local' explanation - in terms of the individual particles in the medium (or segments of the medium), that move only due to the interactions with their neighbors?
In waves framework a local explanation would be that the effect of each force acting on a particle is independent of other forces acting on the same particle (or other particles in general). In your example this explains precisely why the center particle doesn't move: it is experiencing equal opposite forces on itself, one coming from the right wave and one coming from the other.
One final remark: the requisite that the effect of each force acting on a particle is independent of other forces acting on the same particle is precisely the superposition principle. It's not just a global property, it's a local one. It must hold in each point of the medium in order to hold globally.
I hope this animation helps you to visualize the importance of superposition.
The top plot shows a wave travelling to the right, the middle one shows a wave travelling to the left, identical but of opposed sign, while the lower figure shows the sum of the two waves.
Link to gif animation
The same but with different amplitudes
Link to second gif animation
Spring-mass model
Directly from wikipedia:
The wave equation in the one-dimensional case can be derived from Hooke's Law in the following way: Imagine an array of little weights of mass m interconnected with massless springs of length h . The springs have a spring constant of k:
Here the dependent variable u(x) measures the distance from the equilibrium of the mass situated at x, so that u(x) essentially measures the magnitude of a disturbance (i.e. stress) that is traveling in an elastic material. The forces exerted on the mass m at the location x+h are:
This could be the key point: if you look at $F_{Newton}$ as the effect on the central particle caused by waves passing by, you see need to attribute the cause to $F_{Hooke}$, the elastic force. This effect is precisely linear: how do you tell if the resulting force is caused by a single wave of a certain amplitude, or two waves with different amplitudes that sum to the same amount, or infinite waves that again sum to that total force. You simply can't. There's an infinite number of ways to cause that exact amount of force on the central particle.
Final edit: from the animation it's actually not that clear why the wave shouldn't disappear. It is because you are just looking at the deformation. But it doesn't hold all the information: it's a system evolving in time. You have to also look at speed and force at any instant. This final animation should evaporate all your dilemmas: 
You see that when the waves encounter, speeds and forces add, not elide. The elision you see in the deformation domain is just a "coincidence".
Mathematically
The waves are solutions of the wave equation:
$$\Delta f - \frac{1}{v^2}\frac{\partial^2 f}{\partial t^2} = 0$$
This equation can be solved by many tools. The most elegant method is probably Fourier transformation; it allows us to separate the solution in coordinates (that's useful in some physics applications). The solution you mentioned $f = g(z-v/t) + h(z+v/t)$ is only one solution, but it doesn't cover the whole picture (understand the whole space of solutions).
Physically
We can find wave equation in few basics examples. The first one could be the string vibrating for small initial deviations. The other wave equation can be found in Maxwell equation for field $\vec E, \vec B$ or for scalar and vector potentials $\phi, \vec A$. It only means that these waves are physical, but these waves must still satisfy Maxwell equations.
In physics, this is only a special case of what we call waves. We have Klein-Gordon waves, Dirac waves (you will learn this in quantum field theory) or very simple equation from electrodynamics: waves in conductors. All these waves don't satisfy the wave equation in the classical meaning of \eqref{A}. But we can still call it waves.
So mathematically waves are strictly solutions of the wave equation \eqref{A}, physically we call disturbance in the space that are time-dependent or physical entities carrying information that is propagating from one place to another.
The word "wave" has its origin. Mathematicians in history (in post-renessaince era mostly) were working on the description of the musical instruments. So the first origin of waves was from mathematicians studying physical nature. I can recommend you this article [*].
Edit:
When Maxwell equations are reduced into wave equations, it really means that electromagnetic waves exists. That is because of existence of $\vec E, \vec B$. These fields exists, they carry momentum and energy. So if they are solution of the wave equation, they must be waves.
This led many of physicists to think that there is an aether, which is a medium where electromagnetic waves (as light) can be disturbance. And this is the problem known in special relativity when Michelson and Moorley proved that aether (even if it exists) is not important for electromagnetic waves. This experiment was proof for Einstein special relativity and it started the modern era of the physics.
[*] https://www.jstor.org/stable/41134001?seq=1#page_scan_tab_contents
Best Answer
The answer to the first part of my question is on wiki, so I will not answer that here.
After some research I have come up with the following answer to the second part of my questions. There are 4 types of waves we need to consider:
The IPW are basically the 'normal modes' of the infinite string. They are the only type that can be superimposed to form any TW. IPW can also be superimposed to form the other two types of wave (SW and CW) (in a given region where a finite string lies). SW's can be superimposed to form CW only but not TW. All of this information can be summarised on the following diagram.
Sources:
3.http://www.colorado.edu/physics/EducationIssues/baily/courses/BailySP12_GroupVelocity.pdf