When exploring deep questions in physics, like you are, it is important to remember that nature appears to obey the laws of physics. Empirical studies such as science cannot prove ontologically that nature does obey the laws of physics, or obey laws at all. For an extreme test case, consider the concept of idealism. In idealism, one claims that there is no such thing as "matter," only a mental substance that forms the fabric of consciousness. Matter is an illusion, under idealism, brought forth by the shared experiences of those consciousnesses. It sounds a bit absurd, but if you dig into philosophy, you find it is remarkably difficult to find an empirical way to disprove it. People have been trying for centuries. And, obviously, if matter is only an illusion, so is the idea that matter obeys any laws at all!
Coming back from that extreme, even the idealists have to admit that the laws of physics are remarkably effective. As several other answers have mentioned, we know of no single experiment which ever demonstrated a violation of the conservation of energy nor momentum. In fact, there's only one point in all of spacetime where we even postulate that conservation of energy might not have been conserved: the big bang. Even in our study of black holes, we build our equations around the assumption that energy and momentum are conserved.
So does that mean the laws are true? Well, it doesn't prove it. We have gobs of evidence in support of the claim that it is true, but evidence does not mean proof. In philosophy, there is actually a tool to deal with this called abduction. We are all familiar with deduction, a. la. Sherlock Homes, and we are all familiar with induction. A third -duction is adbduction. In abduction, one states that the most likely hypothesis is indeed the true one. Based on this, it would be reasonable to use abductive reasoning to justify the claim that "the laws of conservation of energy and momentum are indeed true." However, abduction is a bit of a sticky wicket when it comes to philosophy. It turns out that trying to pen an exact definition of abduction leads to all sorts of strange behaviors. I believe this is why it takes a back seat to deduction and induction, which people are more comfortable with.
In this sense, we can treat the conservation of momentum and energy as hypotheses. Whether we accept them using abduction is up to us. However, no matter what approach we use, we must admit that these hypotheses have remarkable amounts of evidence behind them. So much evidence, in fact, that most people simply accept them as law.
And perhaps they are law. If one wants to assume that physics applies to "everything," one has to be careful with abduction. However, there is another approach which may be valid. One might define the physical world to be the part of "everything" which obeys the laws of physics, such as the conservation of energy and momentum. In such a case, we have now defined the physical world to be a subset of "everything." If indeed the physical world is "everything," and there is nothing beyond, then the physical world is equal to "everything" (which makes it a subset, though not a strict-subset). On the other hand, if there is "something more," then we know the physical world is a strict subset of "everything," and that physics does not specify the interactions between that subset and "something else."
We could see this in the form of dualism, the belief that there is both physical matter and some mental substance. Dualism is very popular, both because it describes our experiences very well and because it plays nicely with major religions. In dualism, the mental substance is not bound by the laws of physics. There is nothing in physics to suggest that this mental substance, unbound by physical laws, couldn't interact with the physical world, adding energy or momentum. In fact, a great deal of philosophy has gone into trying to find ways to describe that magical process!
So all of that shows the limits of our "laws of conservation." We cannot prove them. That being said, you will not find me relying on the existence of something that rejects those laws any time soon without a commensurate cause to spur me in that direction. The quantity of evidence is just so large. Hypothesizing the existence of something which refutes those laws is one thing; depending on said existence is another.