Disclaimer: My science levels are all over the place so this may not be 100% accurate. I'm specifically concerned about my first paragraph which is a alternative interpretation of Gödel's Incompleteness Theorems. It takes a different view on the matter, one that I feel is correct but am not certain of. I added a second concrete example to illustrate the thought process. Slightly unrelated is Turing's Proof which rankles me and may be related to decoding the universe. See Note2 for an argument on Turing's Proof that I way more certain of than I am of my argument on Gödel.
Our current math theorems only tells us that we can never have a complete system in the sense that we can never know with certainty that our axioms and such are complete, etc.
But this rides on the assumption that the parts our math doesn't allow us to grasp aren't the ones that override this assumption. (Our current axiom set roughly says "You can't have all the rules because you can't logically get to some of them, no matter what set of axioms you pick" but there may be a rule that we have to hit by chance or take on faith, since we can't get to it logically by our current axiom set per Gödel, that translates out to roughly: "Disregard all other rules. The last axiom of math is the 5th item on your shopping list on every billionth day. Just take my word for it."). Like all science we may discover that we need to rewrite the axioms when we hit new areas of math. This could open the way to us discovering the true set of axioms that explains all (if they exist). That is, by virtue of logic failing we could potential be forced to have our minds wander until we hit the right answer by chance (which sounds a lot like the current state of affairs for our current search for a ToE).
A good example for the possible last axiom (well, maybe for a moral system and not a mathematical one) would be the rule: "Everything is a grey-area of some extent, there are no black and white rules." which seems decent if you can apply it to just about everything. But when you get around applying that rule to itself it becomes a statement of "Sometimes things are black and white, because this rule is a grey-area itself." So do you edit it to "Everything is grey except for this rule." or do you take it as it stands or do you toss it completely. The right course of action seems difficult. Assuming that this is one of the true axioms of the universe the two framings have different outcomes: strict rules with a loophole written in (that may or may not be exploited by the universe at any given moment), or with exceptions only for themselves. Either way its status as an axiom relegates it to the "take it on faith" principle which in the case of this self-paradoxical rule is a little harder to swallow that 1+1=2.
So even if the universe is deterministic to a fault we would most likely need a lucky guess or a change in mathematics to get all the rules right.
It could be that even without perfect rules we could move towards a snapshot of the universe at time T. Our certainty would never reach 100% but we could get very very close. If your predicting the future accurately left and right though, then you pretty much don't care if you're 100% right, you might just assume you are spot-on which is basically what Bayesian statistics would tell you. Sure you missed that thing-a-ma-what's-it axiom, but that only causes a simulation glitch once every billion years... Do you care?
If the universe is boot-strappable from a small set of data and has implicit T states, then we could operate on all of this data and recreate a time-location-state(which is what you're asking, so this is the caveat). The simulation could even reduce as far as a random seed number. This may be a number with only finite possibilities. We could test the parameters until we reached one that modeled our current T value correctly. This would get us to the point of modeling the universe without violating the Heisenberg Uncertainty Principle or any other building blocks.
If the universe fails any of these assumptions (is not deterministic, etc.), then our math can give us possible futures with error measures. You could extract information as long as your signal-to-noise ratio for the universe was relatively decent. Non-implicit universes would have diminishing returns on the signal over time and increased noise (which would model our heat as information loss perfectly!).
Note: Figured I'd add that our current science makes the needed assumptions when it assumes to know the effects of the constants on the universe. "Our Constants are so finely tuned it must be the work of an intelligent design". What gets more in the way is the problem of storing a given snapshot of the universe so that we can compute on it. If there isn't a small boot-strappable data set then it can't be done. There are various physical effects that operate as a whole and not additively, so you would need an atom to record all the information about that atom, etc. resulting in the universe. Otherwise you'd need a frequent snapshot of the area of interest to work off of. If you could acquire a frequent snapshot of an area there becomes little need to forecast very far ahead which reduces the error we can have in T+1 allowing more inaccurate rules. (Our weather forecasting is an example. Snapshots of poor detail are acquired and a simulation plots out 7 days. Days closer to the start of the simulation are more accurate. Increasing resolution, simulation frequency, or computing power increases accuracy overall).
Note2: So this isn't directly related except for the fact that if I'm right all of a sudden we can compute Chaitin's Constant which will probably give us a decent platform to create our boot-strappable data set (it also flirts with violating Gödel's Incompleteness Theorems)... So Turing's Proof has always rankled me because it amounts to saying "If you give me a program H that computes Halt() I can give you a program B that always invalidates H" the problem is that his formulation involves B containing H which amounts to saying "I can construct a wrapper function/super-set to H that invalidates it" so the implicit assumption is B is always bigger than H. You can in fact (I wrote a proof) construct an H that uses B to expand itself recursively. This makes sure H is always able to get the correct solution for Halt() as long as H is at-least 1 bit larger than B. If you extend this out to infinity then technically Turing's Proof is right because both H and B will end up at the same infinite size and we can't decide when their the same size (at-least my function can't, although it would be possible if you made a quantum extension of my algorithm (but that may give B new tricks of subverting H as written)) so we end up with an infinitesimal fraction of all programs that are undecidable. But that just means your constant has an infinitesimal error and so is still usable for our purposes.
Edit2: Oh yeah your questions...
- How precise of an estimation could we get of the future? It depends, see the other answers.
- Can a weather forecast for the next day be 100% reliable? I'm pretty sure we can already do this, we just don't run them every day. It's run the minimum amount of times for tolerable accuracy because weather "people"/stations pool money to run a simulation on a supercomputer (or they did anyway...)
- Can we answer questions like "if hitler had died as a kid" with 100% reliability? If we achieve exactly 99.999% accuracy in our rule set for time-steps of 1 yr (noting we'll never reach 100% unless we just take it on faith and it turns out to actually be our magical rule set) and get a perfect snapshot today (2014) then your looking at 99.874% accuracy on our Hitler predictions. So no, not 100% at that point but pretty damn close.
- How much further could we predict from T, with good accuracy, knowing that there is randomness? Well if you compute Chaitin's Constant for your universe you can actually predict your randomness given enough bits of information (this is exactly the same as predicting the seed of a random number generator given x bits where Chaitin's is your seed and the rule set is the generator. You tweak things snipped long explanation until your correct and then randomness doesn't matter). If we take the other track and assume you can't then you have the same situation as for the Hitler situation. The perfection of your rule set determines how close you can get. You multiply the accuracy every time-step (so starting with .999999999999 you lose 1 off the least significant digit each step and there are Z steps each second which is at least the speed of light divided by the number of planck lengths in a meter in magnitude (so 1.855492(18) x10^43) which means for an accuracy to the 12 decimal place you lose at least (.999999999999 accuracy to the 2 x10^43 ticks/sec to the 31600800 sec/yr) .2% accuracy each year (I think I did that right...)). So if we place our limit for good accuracy at 90% then we have 50 years.