Will Determinism be ever possible? What are the main problems that we need to solve to prove Laplace's determinism correct and overcome the Uncertainty principle?
 A: Laplace's determinism is fine if you do no longer consider x(t) as the function you want to deterministically describe but accept the fact that only the wave function $\psi(x,t)$ is deterministic - and even that only when you neglect QFT effects.
A: Experiments such as the double-slit, EPR and Bell inequality experiments show that quantum mechanics is not just a computational tool, but a genuine property of nature. In this sense, Heisenberg's Uncertainty Principle cannot be overcome and there will be no reversal to determinism.
A: Laplace's determinism is not physically correct over long periods of time.   That is, it neglects chaos/"sensitive dependence on initial conditions"/exponential growth of microscopic perturbations already in Newtonian dynamics, which was seriously thought about only in the 20th century.   Being true, this also will not be overcome.   Stochasticity enters some classical dynamical paths with time.
There is subtlety here.    In classical mechanics, or the evolution of the wave function, there is a kind of microdeterminism, so that what occurs in the next instant is fully determined by what occurred up until that point.    It is in the longer time evolution of a chaotic system that stochasticity creeps in.
By the way, Lapalace said
"We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow."   This part remains true in chaotic classical mechanics.  
However, he then continued "An intelligence knowing all the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes. The perfection that the human mind has been able to give to astronomy affords but a feeble outline of such an intelligence. (Laplace 1820)"  This is the part that classical chaos invalidates.
You might also read http://plato.stanford.edu/entries/determinism-causal/ 
Finally, there are also questions whether, in light of general relativity, black holes, etc, we can even speak of a "state of the universe as a whole"    There may not be such a god's eye view altogether.   These issues need a philosophy forum, however.  
A: If you want a minority's view on this subject, here it is:
Determinism means that one asks for a theory that describes unambiguously what is going on, without even the slightest amount of fuzziness. Of course, fuzziness may come in at a later stage, when we inevitably are confronted with the fact that we do not know exactly how nature's laws work under all circumstances, and also we do not know with infinite precision what state the universe is in at any moment in time. But in principle, the theory should be razor sharp. The theory should only talk about realities, things being there with certainty, or not, also with certainty.
The conclusion of most investigators that there is no reality because we are unable, at present, to describe it, or to measure things with as much accuracy as we would like, is indeed a premature conclusion. It is true that all attempts to attribute all of today's observations on nature's laws to something that is happening 'with certainty', have failed, but this does not mean that one should give up. The statement you often hear, that quantum mechanics is a fact, it will stay with us forever, and it means that there will never be a deterministic theory, is premature because that discussion is still far from closed.
It so happened, that a very convincing argument was put up in the 1960s by John Bell. He "proved" that the outcomes of some cleverly designed experiments, as predicted by quantum mechanics and indeed later confirmed experimentally, cannot possibly be reproduced in theories that assume the existence of "reality". The experiments themselves are of course "real", but you can ask what the outcome would be if, during the experiment, suddenly the setting was modified. The modification, it was said, cannot affect the outcome of measurements done sufficiently far away, and then, a contradiction is arrived at.
Does this mean that "reality" does not exist? According to most who investigated this, this is indeed the case, but I do not agree. "Counterfactual" observations are observations that actually are not made, because they are standing in the way of observations that can be made. To demand that these must have a well-defined outcome because otherwise your theory does not describe reality, is debatable. In fact, I now know for certain that this is incorrect, but my arguments fail to come across. 
Anyway, my claim is that theories can be made that exclusively describe reality without even the slightest amount of fuzziness, but only if you understand how they relate to the world that we are accustomed to. You do have to give up a lot: neither atoms, nor waves, not even quantized fields, can be used to describe reality; instead, it is the most minute pieces of information roaming about in gigantic numbers in the universe, that do describe reality. 
Today's understanding of the physical world is not adequate to grasp this situation entirely. In the mean time, the theory of quantum mechanics, as it is today, stands out as a thing of beauty, enabling us better than anything else to describe the behavior of atoms, electrons and the other subatomic particles, as accurately as possible, but with limitations that are closely related to our limited IQ. There's fuzziness, but whether this fuzziness will be there forever, only the future can tell.
A: Determinism as more to worry about stochastic and chaotic processes than with the Heisenberg's uncertainty principle.
As a simple proof, you can easily consider a purely classical system and observe that you don't have determinism as Laplace had hopped.
A: There are no problems we can solve to overcome the uncertainty principle. We can't overcome it. Because it is like 1 + 1 = 2. Its mathematical fact and has got nothing to do with physics.
General uncertainty relation between two operators $\hat{A},\hat{B}$ is
$\sigma_A \sigma_B \geq \frac{|\langle [ \hat{A}  \hat{B}] \rangle|}{2} $
when you plug in position $\hat{x}$ and momentum $\hat{p}$ operator in this equation,
we get 
$\sigma_x \sigma_p \geq \frac{\hbar}{2}$
