Correct me if I'm wrong but my understanding is that newer theories of physics do not exactly overthrow older ones. The conservation of energy for example still holds true for everything that Newton observed it at but for in other situations it falls apart. So E=mc^2 is an extension of the original law. Same thing with the theories of relativity, they explain Newtons observations as a subsection of a bigger system.
Afaik nothing in current physics prevents FTL from being logically possible. Nothing predicts it either but that doesn't mean it's impossible.
General relativity says that maybe you can get something equivalent by taking non-linear paths through space-time but those sorts of things require very radical types of matter to avoid collapsing into a black hole. And realistically, it's IMO very unlikely that these sorts of things would remain in a more complete (e.g. quantum) theory of gravity.
Afaik nothing in current physics prevents FTL from being logically possible. Nothing predicts it either but that doesn't mean it's impossible.