I'm asking apart of the famous twins-paradox of special relativity theory and the ladder-paradox which have been resolved in the literature if there are still some paradoxes or variations of old ones as thought experiments or maybe also some experimental results that today still resist a solution and apparently violate causality?
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No, there are no unresolved paradoxes in special relativity. It is more than a century old and has been thoroughly studied. At this point in history any claim that there is a logical flaw in SR should be treated with the same or even more skepticism as a claim that there is a logical flaw in the foundations of arithmetic. I use the analogy deliberately: Kurt Goedel, prompted by questions from David Hilbert, did find a "flaw" of sorts in the foundations of arithmetic (the famous incompleteness theorem). Both Goedel and Hilbert studied relativity extensively and did not find any such flaw in relativity.
Not only has it passed logical scrutiny, relativity has passed experimental scrutiny. All of modern physics is based on relativity and quantum mechanics. The global positioning system, modern computers, lasers... they all work as predicted by QM and SR. Atomic clocks are sensitive enough to directly measure gravitational and kinematic time dilation predicted by relativity. Time really is relative, and experiments prove it.
Eric Smith
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Perhaps "paradox" is the wrong word for these. "Example" might have been better.
The universe is approximately classical at low speeds, but we are so used to it that anything else seems to be impossible.
The basis of special relativity is that the speed of light is constant. From this we derive the failure of simultaneity, time dilation, length contraction, and all the other features of SR. These all are how the universe actually behaves (OK, they are a better approximation than classical physics.)
So the paradoxes are just counter intuitive examples of how the universe really works. They were invented as teaching examples.
There are other examples that violate our classical expectations of how things should be. Some of them are not completely explained.
For example, see What is time, does it flow, and if so what defines its direction?. This says that time is very different than you would expect. The common sense view of time is that it flows. The present is all that exists. The future hasn't happened yet. The past is over and gone. But the failure of simultaneity implies the Block Universe is how time really works. A succession of events do not come into existence and disappear. The whole block of events in all of space-time just statically exists. That way I can see a distant event as simultaneous with my "now", and you can see it as simultaneous with my past.
But if time does not flow, why do we see it moving forward? There are still questions about the arrow of time. We accept that time does move forward, do physics with it, and it all works. But it is unsatisfying to classical intuition. And even though asking why the universe works this way has no answer from physics, people still do ask.
mmesser314
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Minkowski space is a model for special relativity. A theory with a model cannot be inconsistent.
Moreover, this particular model is so simple (it is basically the study of one particular real quadratic form) that any apparent inconsistency (as opposed to an actual inconsistency, of which there are none) can be resolved quite easily, usually with just a few lines of linear algebra.
WillO
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