Your book is not so out of date as you may think-- The fundamental aspect of nature which is know as "charge" (i.e. the property of electrons and protons which give rise to electromagnetic forces) is not completely understood. In a sense, questions about "why" something happens always seem to involve an answer of "how" a process occurs at a finer level of examination. So, for example, you might ask why a prism disperses the colors of white light-- this can be answered in terms of the properties of light (electromagnetic wave) propagation in a solid, composed of charged particles. While the model you derive will accurately determine the light dispersion you measure, it is makes another set of assumptions (i.e. the activity of charges and fields). Physics, for the last hundred years or so, has attempted to reduce the number of assumptions about the universe that must be made, by building ever finer, more comprehensive models. Unfortunately, the question you ask is on the current frontier of physics. In other words, there are not yet generally accepted models which predict charge (and other quantum numbers such as charm, color, parity, etc.) from more fundamental processes.
I can describe a simple model: Charge is the property of a particle which allows it to interact with light (i.e. photons). An electric field is a model to predict the force which can be observed between two charged particles. The two particles exchange momentum by exchanging virtual photons. (A virtual photon is one that is created and destroyed between local events so that energy is conserved in the long run). The properties of the field, i.e. the delay time before a disturbance is felt from one
particle to the other, the magnitude and direction of the force, can all be predicted from this model which is called quantum electrodynamics. (There is a nice introductory book on this subject by R. P. Feynmann called: "QED - The Strange Theory of Light and Matter" which may be in your library or is about $10 -- the book is written for the general reader).

If two particles are charged, and if one is moved suddenly, -- given that the force is mediated by virtual photons, how long does it take before the other particle feels the change?

It is still the case that at the most fundamental level we don't understand why electric charges behave the way they do. That's also true for gravity and the nuclear forces. Most of the work that has been done has been to characterize the behavior of these forces. There isn't a good theoretical explanation for the origin of these forces yet. There may never be.

I'm not sure I can answer this very well, as I am not an expert on the latest theories in this area of physics. One of the nice things about being at the university is that, whatever the subject, you can usually find someone who knows a lot about it. What I can tell from talking to other people here is that the question has not been answered
in the eight years since your physics book was written, and I'm not sure it can be answered at all.
The theory of how charged bodies interact is very well developed. Using this theory, which is called electrodynamics, one can accurately describe the effects of charged bodies on each other. However, electrodynamics only describes the forces, it does not say why the forces exist. That's the short answer.
If that sounds unsatisfying, I can tell you a little about how the electric force is described, although I can't answer why the force is there. At the microscopic level, physicists like to think of interactions between charged particles in terms of the electric and magnetic fields. Fundamentally, the fields are made up of photons, which are the small bundles of energy which make up light. Charged particles (electrons, for example) are capable of emitting and absorbing photons. One charged particle may emit a photon, and another charged particle may absorb it, and this exchange appears as a force between the particles. (By the way, this photon cannot be observed!) This strange way of thinking about the interactions between charged particles is actually very useful, and lets us make some predictions which agree amazingly well with laboratory experiments. However, as far as I can tell, this description is just another way of saying that charged particles exert forces on each other.
There are theoretical physicists working on a new theory, called "string theory," which may turn out to provide a deeper level of understanding than I have described. However, string theory is not yet complete, and has yet to be tested experimentally, so we'll have to wait a little longer to see what it can tell us.
I'd like to add that most physicists don't worry about why charged particles exert forces on each other. It's often enough to know how to describe the electric force. For example, say you want to understand why metals conduct electricity. We know that a metal is made up of many, many atoms. Each atom is made of electrons and nuclei,
which are charged particles, and you already know what happens when charged particles exert forces on each other. So all of the basic ingredients are well understood. To understand why charged particles exert forces on each other would indeed be interesting, but it wouldn't help you understand why metals conduct electricity. By the way, it turns out that the huge numbers of charged particles present in the metal make it challenging to figure out what happens. The study of systems of many interacting particles is another big research area in physics now.