Each vector represents the strength and direction of the field. Red for the electric field, blue for the other one. Only the base of the arrow counts for the physical location: that is, the field at the point in space where the base of the arrow is (which in this diagram is always some point on the $x$-axis) has direction the direction of the arrow and strength the length of the arrow.
Ummm...your diagram is not labelled with any axes. I assume it is the usual $x, y, z$.
As already pointed out, this is all for one time-value only, changes in time are not being illustrated, you'd need a movie for that (or a good imagination, imagine yourself travelling in the opposite direction to the motion of the wave, at the speed of light).
It is also only good for that one line, but they could've done the same thing for all other points in space, the obstacle is merely practical: the diagram would be a mass of red and blue ink and you couldn't see anything...
B) the wave does its own propagatin, and there is no medium.
C) The wave is a separate issue from any particle. Ummm, what particle? You didn't mention any particle at first. A wave can propagate all by itself, there doesn't have to be any particle.
D) Your second diagram uses a different system. That system is usually reserved for a static field. That is the only way you can have an electric field without a magnetic field, since if the electric field changes, it produces a magnetic field. That is not the only difference. Those are lines of force, not vectors. (They are not vectors with infinite length, ja ja ja..) This system of representing a force field shows you the direction of the force field vector at any point: just pick your point you want to ask about, see which line it is on (not all the lines are drawn but you can interpolate visually in your imagination, of course, just as you had to with the other diagram anyway), and the directin the line goes in tells you the direction of the electric force field vector at that point. But not its length. The length (or strength) of the electric field is given by the density of the lines in that immediate region. Now you see what a different system this is, but you also see it has its advantages.
Okay, by now I hope you see that in the first diagram, the lines are not the field created by a particle, although they are in your second diagram.
I also hope you now see that it is false to say
« The wave is propagating though an electromagnetic field.»
The field is given by that diagram, it propagates in empty space, or in filled-up space, makes no difference. Whether it is a wave or not is partly a matter of taste: if it looks, over time, as if it was bobbing up and down like an ocean wave, you can call it a wave for that reason. Or if, more abstractly, you notice the potential functions satisfy the wave equation, you can call it a wave for that abstract reason. The wave is not something different from the field as it changes over time. The field is the wave, and vice versa.
E) Yes, a charged particle creates a field in all of space, but not that field, not the one you drew, unless the particle is moving or did some moving (it could then stop).
Now since we cannot in fact see the motion in time of the field from your diagram, which is only one time-slice, so to speak, I cannot answer very precisely, nor can you. The relation between the motion of a charge and the propagation of a field can be rather complicated and indirect, it depends on the acceleration of the charge more than anything else. So the very direct connection you ask about is sometimes, but not usually, true.
Beginning to answer your further questions. I am convinced there is something slightly wrong with the diagram. The two charges labelled q form what is called a dipole. A moving dipole radiates, but it produces a spherical wave, with a very complicated structure. But if you consider only the x-axis in space, then this diagram is only slightly wrong in that the amplitude of the field should decay the further out it gets.
This looks like a slight adaptation of the common diagram of a plane wave. The reality or accuracy of the notion of a plane wave was addressed in another answer to one of your other questions. It could be produced by an infinite number of oscillating charges and is a very useful approximation. It is also simpler to understand than more realistic waves and that is why it appears so often in beginning texts. From a theoretical point of view, it is of fundamental importance because every possible wave can be analysed as a linear combination of plane waves. A spherical wave is next in importance, and the analysis of a spherical wave as a linear combination of plane waves is a standard exercise.
But if the x-axis in this diagram is time and not spatial, then the diagram shows the
oscillation of a wave at a point. So you cannot tell from one point whether it is a plane wave, or a spherical wave, has constant amplitude in space, or decays as it should from a dipole. So if this is the meaning of the diagram, it is completely accurate: it shows a sinusoidal wave, and the oscillating dipole produces a sinusoidal spherical wave if the charges oscillate sinusoidally. But even if this is the intention of the artist of the diagram, the label `k' is then incorrect, since that means the wave vector which indicates the direction of propagation of a plane wave (usually, it is not usually used for spherical waves although it could be). More later.
For a plane wave, the concept of direction of propagation is easy, and usually the diagram you have is for a plane wave. But there seems to be a dipole here, and for dipole radiation there is no one « direction » in which the wave propagates, nor is the electric field vector always in the same direction. But along the axis that is here labelled with `k', perpendicular to the axis of the dipole, the E vector is always parallel to the vector connecting the two elements of the dipole, as shown in the diagram. But in other locations in space, the E vector does not have to have that direction.
This diagram is an incorrect mixture of the idea of a plane wave and the idea of
a dipole radiating a wave. If the x-axis is spatial and the dipole is oscillating sinusoidally then the diagram only needs two slight corrections: the amplitude of the wave should be decreasing roughly speaking as $1/x$ (there are also terms which decrease faster), and the $k$ should be omitted from the labels, since for dipole radiation there is basically spherical propagation, not planar propagation with one direction.
IN particular, the wave also propagates « up » in the z direction the axis along which the dipole is oscillating up and down. Its amplitude is weaker, it falls off as $1/r^2$, but the E field here is in the z direction, parallel to the dipole axis, and in the direction of propagation, which is what you were asking about. So it can happen, just not for plane waves. The wave produced by a dipole is more like a spherical wave than it is like a plane wave, and this diagram hides that fact.