- Our best model of the Universal microstructure is in fact a discrete one, and not continuous as you suggest. Everything is quantised, even space and time. Due to quantum uncertainty, distances below the Planck length and time periods shorter than the Planck time cannot occur. The continuous equations which we use to model larger-scale events break down at this smallest scale. Loop quantum gravity is constructed on this principle and is a strong competitor to string theories. It is also worth noting that discrete does not imply binary. Even an analogue computer is discrete in the sense that it resolves down to the nearest quantum of signal energy and no further.
- and 3. While not all functions can be computed exactly, they can be iterated indefinitely - which is what the Universe appears to be doing. Programming tricks are used to keep diverging functions within manageable limits. (tricks which my first pocket calculator intriguingly lacked). Phenomena such as the uncertainty principle and renormalisation could be examples of such computational tricks.
Posit a sufficiently powerful intelligence with vast resources in a higher-dimensional universe, and why should their unimaginably sophisticated computers not be able to simulate ours? (See at least one other answer)
And there, of course, lies the weakness of the whole naive edifice; despite all the above, your uneasiness is well founded. It is merely dressing up Bishop Berkeley's idealist God in SF technobabble. For "the mind of God" read "a super-alien super-computer". Why bother with the computer? Berkeley's God did not delegate the Creation to the Archangel Gabriel, He just got on with applying Occam's razor, as any good super-alien would do.
It is true that modern information theory is playing an increasingly fundamental role in thermodynamics and cosmology, to the point that at least one respected physicist has remarked that, "The Universe begins to look more like a great thought than a great machine". But to make that leap, and say that the Universe therefore equates to information, begs the question as to why the most complex structures in the Universe (our heads) are so full of illusions, mistakes, paradoxes, contradictions, fantasies and outright lies, yet none of that information manifests itself physically.
Just another example of Einstein's old saw that philosophers may make bad scientists, but scientists make even worse philosophers.