An Alcubierre warp drive[1] would be capable of enacting causality faster than light, no?
Imagine that human civilization survives and progresses for 100 million years. Do you think that they'll be able to look back on any of our "laws" without laughing?
At that point we'd likely be far beyond Dyson spheres, harvesting large quantities of antimatter, solar system scale particle accelerators, artificial black holes, quantum supercomputers, etc., etc. Whatever would be discovered is almost guaranteed to be unimaginable by us.
For the artificial black holes, Dyson spheres, antimatter, quantum computers, and giant accelerators to actually work, the laws of physics they're based on can't be that wrong. They can be incomplete, requiring their place within a greater theory, but not simply wrong. There is observation consistent with the notion that the speed of light has been constant for the last ~14 billion years, everywhere we can see.
There is no theoretical limit to the rate of metric expansion, but unfortunately metric expansion doesn’t allow for FTL communication. Let’s put aside issues of lightspeed in an expanding universe though because we don’t even need to go there to address this basic issue of why you can’t have a brain past certain limits.
At some point limited by the speed of light even in a static universe, your “thoughts“ would be incredibly slow. What would it mean to have a neuron the width of the galaxy? Would you be particularly intelligent just because you had a lot of individual parts from a certain distance? There’s probably some theoretical upper bound on the optimum size for a computing body in a universe bound by c, based on the total amount of entropy which could ever be contained in a given region and the rate of signal transmission.
As to adjacent dimensions and that kind of thing, who knows? In current physics that would be like magic, and magic can be whatever you want it to be after all.