About faster than light...
I know (in fact I am currently yet studying) different extensions of relativity. Some options naturally arise:
1) Yes, Ben... Sudarshan's (and Recami's) Meta-relativity is one "option", somewhat oldfashioned. Problems: tachyons have not been observed in Nature yet.
metarelativity paper 2012
2) Carlos Castro's/Matej Pavsic extended relativity in C-spaces gives another posibility. You can have different kinds of velocity in C-space. So, the trick used there to avoid the speed of light limit is to add "extra degrees of freedom" living in the C-space. Note that it also keeps the notion of invariant relativity! Problems: we have not discovered apparently any experimental hint of C-space. Relativity in C-spaces: review
3) I discussed another option recently in my blog entry, something quite unknown for many physicists (or neglected) and something that sci-fi writers don't understand. Hypertime. If you have "new" timelike dimensions and different speeds of light, or even if you have multiple times and the same invariant velocity, you can modify the speed of light limit. It is the hypertime and not the hyperspace what makes faster than light motion possible. Note that you can even keep a notion of invariant speed. crystalline relativity
Problem: apparently, Lorentz symmetry holds yet in any experiment, so Kalitzinian metrics (semiriemannian metrics) have not appeared in Nature yet (even the hypothesis of the spacetime quasy-crystal is crazy, but Wilczek himself or Petr Jizba have proposed similar ideas...)
i) Time crystals ii)Time crystals II iii)World crystal
4) The general relativity trick via wormholes (i.e., non trivial topological connections between two points in spacetime) or via Alcubierre warp-drives. Problem: known quantum instabilities and weak energy condition violations.
Some time ago, even something like the Weak Energy Condition was critized (I think there is no such an opposition to WEC now), but I believe the main problem is of course the quantum theory (something I gave up in the original question and that would deserve additional research/thread for any of these 4 options or answers).
What about classicality? With careful analysis, the 4 options above can be considered "classical".
What about causality? Nobody understand what time is and a change in our notion of fundamental symmetry and what means "a clock" should avoid the causality problems. In fact, with multiple time-like dimensions, it is hard to assest if what causality violations cause in 1d time could happen in nd time...And I am not sure how multi-time theories can avoid causality, but I think it could be possible. Note that there have been studies of mechanics with multiple times in the recent literature.