Of course, you are right. An electromagnetic wave (e.g., light or radio waves) must be created by acceleration of charges (in the classical picture, to which I'd like to stick for a while in this thread).
The sources (and the only sources) of the electromagnetic fields are charge and current distributions. If these are time dependent (and not just the Lorentz boost of a static charge distribution to another inertial frame, where they move with a constant velocity) electromagnetic radiation is produced.
These fields are dynamically propagating not only within matter containing charged particles but also in free space. That's what Wikipedia might mean by "self propelling". It's a bad wording, however, because there is nothing "self propelling" here. It's just like a ball you through, and it's moving further without being in contact with you anymore. In free space, i.e., without gravity and air friction, according to Newton's Laws it would just go ahead in uniform motion. It's not self-propelled. In a similar way also the free em. field travels on in free space and is not self-propelled.
It is also a wrong picture you get when it's said that the changing electric field causes a magnetic field and vice versa. You cannot interpret the corresponding homogeneous Maxwell equations (i.e., Faraday's Law and the Ampere-Maxwell Law) in a causal sense. There is only one electromagnetic field. It's even an observer dependent statement what you call "electric" or "magnetic" fields. The electromagnetic field as a whole is a frame-independent concept (described by an antisymmetric 2nd-rank tensor in Minkowski space, called the Faraday tensor).
The "strength of the field" in a certain sense can be associated with it's energy density, which is given (in Heaviside-Lorentz units) by
\epsilon=\frac{1}{2}(\vec{E}^2+\vec{B}^2).
It depends on how vigorously you wiggle the charges, labeled with \pm q in the figure, which is, however, very misleading, if not entirely wrong. What seems to be depicted is the most simple example of a radiation source, the Hertzian dipole (which is realized approximately by, e.g., a linear antenna with a harmonic current; it becomes exact in the limit of vanishing length of such an antenna). The field looks entirely different from the plane wave plotted in the picture. A much better picture (even animated) can be found here:
http://en.wikipedia.org/wiki/Dipole_radiation#Dipole_radiation
The 5th question cannot be answered other than by "that's how nature is". The Maxwell equations which describe electromagnetic phenomena to high accuracy (as long as quantum effects can be neglected) are fundamental laws which have been found after some centuries of observations and just summarize in an elegant mathematical model these observations. They canno be derived from simpler principles and are fundamental in that sense. The existence of electromagnetic wave fields has been in fact predicted by Maxwell and have been found later in experiment by H. Hertz. This is a typical example for the way how science works: You start from a lot of observations, doing quantitative measurements and then find a fundamental law (which is a very rare event, by the way) which leads to further predictions about phenomena that may never have been observed before (or as in the case of electromagnetic radiation have not been realized as such; nowadays we know that visible light is nothing else the electromagnetic radiation in a specific range of wave lengths our eyes are sensitive to). Then you can try to do new experiments to check, whether electromagnetic waves really exist. If not, you have to modify your mathematical model or create a completely new one. If you find them, as is the case here, you have consolidated the validity of the model, and so on.
