The relativistic principle of causality

The notion of "Causality" is notoriously subtle, something philosophers still struggle with and is probably impossible to define rigorously. A search of something like the Stanford Encyclopedia of Philosophy will turn up a dizzying array of articles that all show this notion tests the intellect of very many highly capable and insightful people.

But from the point of view of physics, one can say that it's simply an inductive inference that because, experimentally, certain physical processes are always observed to happen in the same order as time measured by our clock advances, we hypothesize that they will always happen in this order. I've never eaten boiled eggs in my 52 years of existence without my having to boil them first, for example. Notice how this notion depends on known, experimental physics: we have to make observations for each new process to establish whether there is a "preferred" order and what that order is, although this preferred order in physical processes is all around us from the time we are born. There'd barely be a human alive who could not tell when a movie of a glass shattering on the floor is being played backwards.

So to answer one of your basic questions, no, causality is certainly not proven: it's simply an hypothesis begotten of inductive inference. An experimental result.

"Relativistic Causality" is simply the hypothesis that observed preferred orders of physical processes are not disturbed by transformation between inertial frames. There's no reason to believe that well established orders of processes are observer dependent, so we make the hypothesis that they aren't. Not the least because a theory wherein they were would be greatly more complicated!

This is a particularly important point in special relativity, where the notion of simultaneity is observer dependent. The Lorentz transformation, with its conserved signatured metric, gives us a way to make sure that established orders of physical processes are not observer dependent, even though simultaneity is. And that is if make the hypothesis that no two observers can move at speed greater than $c$ relative to one another. Supraluminal boosts do change the time-order of events. So if we make the hypothesis that certain processes happen in a given order and that that this order is never observer dependent, then we are forced to make this no signalling-faster-than light inference; to do otherwise would be to allow the possibilities that gainsay our initial hypothesis. Interestingly, the only other ways that a time and a space dimension can be mixed through linear transformations is by rotation and this is ruled out because one could then always find a boost that would reverse the order of any set of events.

In closing, notice in physics that there is a great deal that doesn't easily fit into this "causality" framework: for very simple systems - like systems of currents and the system of electromagnetic fields that they "source" described by Maxwell's equations - it is very hard if not impossible to ascribe the categories of "causative agent" and "effect" to the system components: current and field are intimately mixed by coupled equations. This is not surprising, because Maxwell's equations by themselves are acausal: any solution can be turned into an equally valid one by inverting the time co-ordinate. Causality is imposed on Maxwellian electrodynamics by hand simply by discarding the so called advanced wave potentials and keeping only the retarded potentials. This situation is true for most of the fundamental laws of physics. However our everyday World and its processes are complicated enough that thermodynamics establishes the preferred orders which we ground our notions of causality on.

Causality in physics is not defined in a satisfactory way. The meaning of cause and effect has been debated for centuries. In the end, at the microscopic scale (with the exception of a few processes) the laws of physics are reversible, and the best you can do is define if two events could have influence on each other. Everything are particles that collide, split or merge, and each of these processes can happen in reverse. Thus there is no much sense of talking of cause and effect at the microscopic level. It does at the macroscopic level though, and it is related with the perceived arrow of time, which it is also debated on how it is originated but is most likely a result of the second law of thermodynamics. It is a statistical law and thus causality is easier to define, and very useful as a concept in many contexts. But while in some situations the interpretation is clear cut (such as the car killed the boy), in others is less clear (the butterfly effect), and in others still less clear (like in a closed time loop, for instance, you are a scientist that travels to the past and clone yourself, so you are born not from parents but from yourself. This last example does not involve any paradox and does not yet contradict any laws of physics. Just my two cents, not a thorough answer.