The relation between entropy and information is well established; indeed, Shannon entropy is the seminal measure of the information in a system.
The other question, about determinacy and information, is more complex, and even more complex yet when extended to the entire universe. Let us set aside, for now, the fact that quantum mechanics would seem to suggest -- under conventional interpretations -- an inherent indeterminacy.
We'll start from classical thermodynamic entropy: 'Maxwell's Demon' is a hypothetical entity with 'perfect knowledge' of a system, capable of sorting molecules according their energy, but Landauer showed that such a creature, in order to function, would itself have to increase the net entropy. Specifically, if there is a logically irreversible change in information, other degrees of freedom will experience an entropy increase. Naively, think of your computer's CPU getting hot. This means that at a minimum, by churning through our propositions in our 'theory of everything' for this determinist universe, we won't be decreasing the entropy, which is OK by the second law: consider an isolated system that has reached thermodynamic equilibrium, and then consider whether you are willing to accept that the universe may be a closed system.
Quantum mechanics complicates it. Here, if conventional thought is followed, there is an inherent randomness. Specifically, if we further accept unitarity (ie the sum of possible probabilities of measurement outcomes is 1, fairly uncontroversial), then we have, thanks to Everett, a proof that the second law is implied (the theorem is independent of his `many worlds' interpretation, which is rather an attempt to explain what the implications are) -- pg 122 here.
What this tells us, in summary, is that classical thermodynamics is consistent with a determinate universe under some constraints, but the very reason that QM says the universe is not determinate is precisely an explanation of the second law. Can we infer from this that QM is the sole cause of this? Under the assumption that QM is a complete theory, then yes. We know that it is isn't, in the sense that it doesn't tell us everything we would like to know about the world -- but most of the attempts to extend it to include general relativity, for example, and certainly the credible ones, accept the same assumption of unitarity, and the uncertainty principle, and so forth. If these theories, such as string theory, don't end up introducing another mechanism by which entropy would increase, independent of QM randomness, then that is indeed the cause.