Part One: The Past
IF we live in a casually deterministic universe, then everything in the past that gave rise to the present really existed. The past must be perfectly preserved as a record of what has come before the present. Every event, all objects and all energy were actually there when the present swept past them. In an infinitesimally small moment, the then-present became a part of history stored in a subsequent present in an ongoing process. Our memories appear to mimic this process--imperfectly. We live in the present, as does the universe. We recall the past, as the universe does. And we predict the future, which the universe also appears to do.
A different question arises: Does the past (and everything in it) still exist? We can infer that it existed by observing the present (the only thing we can observe), but is it a fair statement that it still exists in the present? Anything that can be interacted with can be said to exist. Measuring or observing is a form of interacting. If the past can be detected in evidence in the present then it can be said to continue to exist once it has been left behind by subsequent presents. Whether or not we can interact with the past, a truly deterministic universe ruled by causality must interact with the past or at least maintain it, in order to maintain a coherent present.
The universe is deterministic only if it is entirely deterministic. If anything cannot be related causally to something that came before, then both causation and determinism break down. If there is something in the last moment of the universe that exists only for that final moment and can't be causally related to something that came before, then the universe is not deterministic. The evidence suggests that the universe is deterministic. Quantum Mechanics are probably the best evidence against determinism that have arisen from science to date--IF one defines probability as being insufficiently deterministic. There are two arguments against this: First, One can view quantum wave theory as actually deterministic. If there is no collapse caused by the intervention of an observer, then the wave function describes the entire process. Another argument is that there are laws making probability deterministic in fact, but that we cannot properly discern them. Along those lines is the possibility that there are completely deterministic rules that govern the functions of quanta (as opposed to probability in general) but they are as of yet not known.
Determinism is an absolutist theory. It can only be true if it is absolutely true. If anything can be shown to not be deterministically related to something else then determinism is broken. At the moment, one can reasonably say that determinism appears to break down at the quanta level. The problem is that perfect knowledge of the universe is impossible. That imperfection might always obscure actual, fundamental determinism by making deterministic processes appear random (in the same way that the motions of the planets were seen as random in the age before astronomy). That is a reasonable interpretation of the evidence presented by quantum mechanics and one that preserves actual determinism.