How sound waves and the air work I've come from trying to understand how Ethernet works to studying about waves. 
Question 1
So what I don't understand well is that the medium of waves return to their rest position or equilibrium after the energy leaves them. The article I read said that the particles interact with each other, and that is how waves work. Is this possible without a connection between the particles?
Now let's assume a sound wave. The first air particle is displaced and hits the one next to it. How does it know which place to return? In the website, it said the air particle (or molecule) returns to its rest position as a result of collision, but if we imagine something like the beginning of a pocket ball game, the original ball causing a disturbance only roughly returns near its original position if at all. Also the other balls would be permanently displaced, which is not the characteristics of a wave. 
If the original particle knows where to return due to the void it created, couldn't other particles fill that void also, since the void isn't exactly reserved for one specific particle?
If the situation is such that the table is full of balls so that there is little space between each ball, the balls wouldn't be displaced as much, but still it can hardly be described as "assuming its original position" like a slinky coil does. I haven't tried, but I can't imagine pushing and pulling back a ball repeatedly in and out of a huge group of balls making ripples like sound waves do, or do they?
Question2
Also, if the molecules involved in making the waves can't pull each other, does that mean you can't cause sound waves by a pull force?
Question 3
Is the difference between creating a puff of air and creating a sound wave just how short and abrupt the disturbance is, like when snapping a towel?
 A: This semi-duplicate Sound wave propagation should answer your questions far better than that which I have have written below. The answer by Floris on this page explains the process of movement of air molecules in terms of pressure variations, with more detail, (and accuracy) than I do here.

Now let's assume a sound wave. The first air particle is displaced and hits the one next to it. How does it know which place to return? In the website, it said the air particle (or molecule) returns to its rest position as a result of collision, but if we imagine something like the beginning of a pocket ball game, the original ball causing a disturbance only roughly returns near its original position if at all. Also the other balls would be permanently displaced, which is not the characteristics of a wave.
If the original particle knows where to return due to the void it created, couldn't other particles fill that void also, since the void isn't exactly reserved for one specific particle?
If the situation is such that the table is full of balls so that there is little space between each ball, the balls wouldn't be displaced as much, but still it can hardly be described as "assuming its original position" like a slinky coil does. I haven't tried, but I can't imagine pushing and pulling back a ball repeatedly in and out of a huge group of balls making ripples like sound waves do, or do they?

I think I have answered most of your points here, or at least tried to. The air molecules don't know anything, I appreciate that you don't mean that literally but what I mean is they don't have any preordained place in the room. There are so many so tightly pushed together that it makes little difference if the original air molecules that started the wave move off somewhere else, there are plenty more to move very quickly, much, much faster than you can speak, and take their place.
Some numbers:
For typical air at room conditions, the average air molecule is moving at about 500 m/s (close to 1000 miles per hour), so they fill any gaps pretty quickly and allow the sound to be carried continously. An average room has approximately $2.5 × 10^{25}$ air molecules per  cubic meter.
Each molecule occupies a volume of $4 × 10^{−26} m^3$ and the spacing between molecules is on the order of the cube root of that, or 3 nm.
So there are a lot of tiny objects jigging  around close to each other to carry the sound very efficiently

Hmm.. I think you have a point in the second paragraph. I can easily imagine being in a pool of water and using my hands, creating a wave by pulling. The water will immediately fill up the space where my hand was, and that wouldn't really be about the water being attached to my hand. But even then, I wouldn't think all the water molecules returned to their original position afterwards

I think you might have preconceived notion of a sound wave being absolutely linear, at least for a few inches in front or your mouth, it's not. It spreads out immediately but we have so much control over our voices in tone and pitch that we automatically adjust when we know we can't be heard. I don't want to concentrate exclusively on speaking, but to me,  it's the easiest way to get across the idea of a continous sound, such as  a conversation  because our lungs are working to keep the sound wave moving across the room, as opposed to a sudden, once off noise.

Why do they return to their orginal position?

The air molecules are  pairs of atoms, nitrogen $N_2$ and oxygen $O_2$, connected by electronic bonds. But it takes far more power than we have when we speak,  to carry the sound by compressing the actual molecules, so it is the spatial movement of the molecule as a whole that conveys the sound. Once we speak, we create a tiny partial vacuum and the air molecules rush in to fill it, so that if the wave is not finished it's compression cycle, there are molecules there to fill the gaps and continue the wave. We  rely totally on the movement, (translation in physics jargon), of air molecules to carry the sound.
They return to roughly their original position because the air ahead of them is compressed and pushes back against them. Try talking into water and it's much denser, somebody just under  under the water surface will see a face making like a goldfish, but they will hear no distinct sounds. The air molecules  also return in part,  because when we speak we pull air back in and that process, although it obviously  diminishes with distance, keeps the wave going. Finally, they return to their positions because there are so many of them, they don't/can't travel very far in the first place.

Is the difference between creating a puff of air and creating a sound wave just how short and abrupt the disturbance is, like when snapping a towel?

Its not exactly the same. When you snap a towel, you send a wave of air out, but this is not the same as speaking. When we speak, we are constantly pushing and pulling air in and out of our lungs, pushing air out to make the sound and pulling air back in to get ready for the next word. So that process is like a spring, pushing your words ahead of it through the air towards the listener.

Also, if the molecules involved in making the waves can't pull each other, does that mean you can't cause sound waves by a pull force?

You create a sound wave by starting a longitudinal compression or rarefaction wave, so it really does not matter if you use a pull or a push force to get the wave going. As an illustration of this, you can easily make distinct words, such as saying the word "car" for example, on the in breath or the more effective out breath. A shout or normal speech obviously is more effective when the air is pushed out, but it is a matter of degree more than it is absolutely impossible to achieve.
I think that in your question, you may not fully appreciate how relatively dense the air in a room is. Every square inch of the ground around you has 15 pounds of weight on it, that is a lot of air molecules available to carry the sound of your neighbors dog  barking all the way from a rear garden or across a park to your ear.
A: In an elastic solid, particles return to their equilibrium position after the sound wave has passed: because they are attached to each other they have a definite "sense of place".
Air molecules don't have a fixed position: the jostle around, and the average effect of their motion is experienced as pressure (number of particles hitting an area multiplied by their change of momentum normal to the surface = force per area = pressure).
Now when you increase density, you increase the number of molecules per unit volume; this increases the number that hit a unit area per unit time, and thus the pressure. But since the molecules "in the next slab of air over" don't have an equally high pressure, more molecules will move from the high pressure area to the low pressure area than vice versa, so the high pressure "wave" moves on. If you lower the density in a region temporarily, the opposite happens: molecules from adjacent higher pressure areas will move in "to fill the void".
All this is possible because, if you imagine a box full of air with an imaginary plane running down the middle, a roughly equal number of molecules will move across the plane from left to right, and from right to left, every second. If you start with one color of gas on one side of the membrane, and another color of gas on the other, very quickly the two will mix - this is called diffusion.
All of which is a long winded way of saying: there is a microscopic picture of molecules moving that explains pressure, diffusion, and sound waves. This picture tells us molecules have no fixed position; but they do have a tendency to move from high pressure to low pressure.
I also recommend reading the question/answer that I wrote previously. It has a possibly helpful picture.
A: It is not really possible to model the propagation of sound in air by calculating the paths of individual molecules. Instead, the speed of sound can be calculated as by Isaac Newton, treating air as a continuum, using density and compressibility. (Ok, one needs the adiabatic compressibility, this is a 20 % correction on Newton's value.) 
The justification for this is that the wavelength of shortest audible sound waves in air (a few centimeters) is much longer than the mean free path at atmospheric pressure (was it 20 micrometers?). So pressure and temperature are meaningful quantities on these length scales.
