Why do positive charges attract negative charges? From school and/or university we know that there is Coulomb's law that allows us to calculate the amount of force between two electrically charged particles.
However, I have never found any information on the origin of this force.
Is there any explanation as to why this force exists? Is it really fundamental (i. e. it is not the result of any other forces like Archimedes force or the friction force for example)? Is there any evidence here? Does it propagate at light speed or another speed and why?
 A: In school, we simplify slightly.  We talk about science as if it tells us the truth about the universe in what philosophers would call an "ontological" sense.  In reality, science is all about making models that explain things and predict things.  We make our models, then turn up the heat, and see if they hold or if new behaviors occur.  Often we find that behaviors we saw before were just special cases of a more generalized pattern.
If a pattern survives us turning up the heat on it as far as we can go without giving way to new and more complex patterns, we call these patterns "fundamental laws."  And they get to keep this terminology until we prove otherwise.  As the most famous example, Newtonian physics was "the laws of kinematics" for several hundred years.  His laws were "fundamental" until the 20th century finally gave us enough capability to peer into the very-very-fast, and discover things that did not obey Newton's laws.  The behaviors that they obeyed were named Specific Relativity and General relativity.  Newton's laws were a special case of this when objects were slow (and by slow, I mean things below a few million miles per hour... the bar is pretty high).
So, in that sense, we currently consider the forces of electromagnetism to be "fundamental."  They appear at the core of our theories, unmolested.
Well, almost.  If you get into quantum mechanics, you find that photons are the "force carrier" of the electromagnetic world.  When you get down to really small scales, these photons are seen as the way electromagnetic forces are transmitted (and, to answer one of your questions, they are why it is said that the effects of electromagnetism propagate at the speed of light).  But we still call it electromagnetism.  It just looks a little different.
Dig further, and you find string theory.  String theory is an open collection of theories which are still not yet proven, but are an effort to simplify things so that they can all be explained via one mechanism.  String theorists would based electromagnetism in the way strings vibrate, so we might argue that string theorists do not consider electromagnetism to be "fundamental."  Of course, its a standing question as to whether any of their theories will ever be proven out (currently they all require particle accelerators with vastly more energy than we have today).
So we call electromagnetism "fundamental."  Its generally a good choice of word.  unless you get into the nether regions of physics, it will indeed be a fundamental force.  But its good to understand what is meant when we say these things.
And for fun, you'll note that I talked about the laws of electromagnetism, not just Columb's law.  There are actually four fundamental equations governing electricity and magnetism known as Maxwell's equations.  Columb's law can be derived from them if you assume point charges generate spherically symmetric fields. And, if you have fun digging into relativity later, you will see that the effects of magnetism can be explained by starting with electricity and accounting for relativistic effects.
A: The electromagnetic interaction is one of the four fundamental forces of the universe. It just so happens that there are two types of charges that interact under this interaction. If you put two particles with the same type of charge together, they repel each other. Two with different types of charge, they attract each other. Nothing in particular pushed us to call them positive and negative, just that the math worked best and simplest that way. Electric charges aren't really "opposite", just different, we named them that way because of the mathematical relationships we found at first.
Down the line, one discovers that the electromagnetic force is mediated at the speed of light by a particle acting as a carrier, called the photon, or the quantum of light. It is the particle responsible of transmitting and mediating all electromagnetic phenomena, such as electrostatic forces, like you asked about, all the way to electromagnetic waves, such as radio waves and light.
All of these phenomena take place at the speed of light, though in classical physics we disregard the lag that takes place to simplify the scenarios we study for simple systems.
