When waves transfer energy by pulling neighbours sideways to the direction of travel, the waves are called transverse waves. In the simulation below you can see energy move to the right while individual particles vibrate up and down about fixed points. Source
Source : Dr. Daniel A. Russell
(Source)
Question 1: Why does moving particles up and down cause energy to move to the right, and not left?
If I moved the particles down first and then up would the energy move to the left?
It is really just a matter of where the motion comes from. Let me explain with an analogy.
Imagine holding a rope at one end, which is fixed at the other end to some mechanism that can measure energy transported by the wave on the rope. You starts shaking the rope, and then energy start flowing from you to the other end of the rope; as you can guess, it is not really a matter of left to right or vice-versa. The energy flow is from its source (in my example, you shaking the rope) through the medium - the rope (in every possible direction). If you held the rope at its center and shook it, energy (and thus the wave) would propagate in both directions.
In water waves (for example the one you see after throwing a pebble in a lake), the energy moves from the pebble outwards, forming circles.
You can also see that, in your example, since every particle is moving up and down in harmonic motion, it doesn't really matters if you start by moving them up or down, and it wouldn't change the direction of energy flow.
Look at your bottom image.
You have a pulse moving from left to right.
What does that pulse consist of?
It consists of particles which are displaced from their equilibrium position and hence have potential energy and particles which are moving and hence have kinetic energy.
So that pulse has energy stored in it, the sum of the potential energy and kinetic energy of the particles.
The pulse is going from left to right and so that stored energy in the pulse is moving from left to right.
Energy is being transported by the moving pulse from left to right.
The top diagram is just a succession of pulses transporting energy from left to right.
So the up and down movement is the origin of the energy being stored in the wave.
The left to right movement of the wave is the energy stored in the wave being transported from left to right.
In the pictures you have, left and right are interchangeable because the pictures are generated using a mathematical formula similar to $$y=A\sin \left[2\pi(f t - \frac{x}{\lambda})\right],$$ where $f$ is the periodic frequency and $\lambda$ is the wavelength. Simply changing the sign of the $x$ term will reverse the direction of the wave. Or, you could turn the screen upside down :).
Real waves can be described as superpositions of pure sinusoidal waves based on something known as the Fourier Theorem. Standing waves, such as those formed on guitar strings, are the result of waves travelling in both directions.
Up and down are relative also. The important idea is that waves initially travel away from an initial disturbance. When they encounter some non-uniformity in the medium, part of the wave can be redirected (refracted and/or reflected) and travel in a different direction. When two differently directed waves (wavelets?) cross each other, then both contribute to what the medium does, but then travel on without change in their basic parameters of frequency, wavelength, or direction. This is called superposition.
The issue is that is fact you are not simply moving a particle up and down. You are moving a particle whose position affects other particles (all around it).
If a bunch of particles do not interact in some way, then moving one has no effect. So in your particle wave, there was to be some interaction between the particles in order to generate an effect that moves across the field of particles.
