Why do we have a universal speed limit? Is there a more fundamental law that tells us why this is?
The more fundamental laws are causality and locality. Causality epresses the fact (or assumption) that effects follow causes, and locality expresses the fact (or assumption) that fundamental causal relations are described by differential equations.
Given these two fundamental principles, the logic of mathematics dictates that the differential equations are either parabolic (heat equation like) or symmetric hyperbolic (wave equation like).
If they are parabolic, there is no speed limit. For example, according to the heat equation, heat propagates instantaneously to arbitrarily far places, though suppressed exponentially with distance.
If they are symmetric hyperbolic, mathematical theory implies a finite propagation speed. For example, this is the case for Maxwell's equations, which limits the speed of electromagnetic signals to a number called the speed of light.
It is an experimental fact that Nature behaves according to the second possibility - even independent of considerations of the speed of light. There is overwhelming evidence that all fundamental processes in Nature are of the symmetric hyperbolic kind. Even heat - the heat equaion is just the simplest approximation, in which the speed limit is lost. But more sophisticated derivations from nonequilibium statistical mechanics produce symmetric hyperbolic equations, which become parabolic only upon further approximation.
That the limiting speed is the speed of light is very likely but not necessarily the case. It is linked to the assumption that photons are massless. If photons were massive but gravitons are massless, the speed of light would be smaller than the theoretical limit of signal speeds in the universe - which would then be the speed of gravity.
However, according to the particle review of the Particle Data Group, the upper bounds on the mass of a photon are extremely tiny, and observations are currently in full agreement with the assumption of massless photons.