The speed of light is actually the speed of causality. It just happens that since photons don't have rest mass, they always move at the speed of causality.

Gluons are also massless like light, but don't exist on their own. If there's a particle that carries gravity, it would also be massless. You could say light travels at the speed of gravity.

This only implies that FTL transport would make causality travel a bit faster... but not subvert it (i.e. "travel through time" or "change the past").

From what I remember reading about relativity, I believe the difference is that speed of sound is just an arbitrary speed of mechanical waves in the air, while speed of light (in a vacuum) is literally what defines what it means for two events to occur before/after each other or simultaneously.

Unfortunately, I am confused about implications of that.

Air is a sufficiently different medium from space-time that comparing them this way is not useful.