Re, chaotic behaviour: I don't think there is any simple similarity. Chaotic systems are computable in that you could in principle keep throwing more bytes and CPU cycles at it to get the error down to any bound you like.

Regarding the first question, I am only repeating pop-science here.

I've read (In something by Roger Penrose, presumably The Emperors New Mind, but maybe in The Road to Reality) that the Maxwell Equations are uncomputable only if the initial conditions are functions that don't have Fourier transforms.

So really it is the field dynamics that are non-computable. I might have been talking nonsense when I the fields themselves were non-computable. But then again, the kinds of functions that don't have a Fourier transforms are also very rough, jaggedy things that probably can't be approximated by any algorithm.

And speaking of rough jaggedy functions, that's what you get if you evolve a chatoic system to infinity. So maybe there is some connection after all, but it is well beyond my understanding.

Chaos Theory is nowadays studies as part of Dynamical Systems Theory.

So that kind of computability might be the mathematical algebraic computability (as in does this monstrosity have a closed form?), which can always be approximated (of course with fast accumulating errors).