I have a Ph.D. in Mathematical Physics, and have thought about these questions for some time. I am going to write down a few of my thoughts as an answer. The basic question is, it seems to me, "How can we reconcile the biblical creation account with its (at least literal) account of a young earth, with scientific evidence for an old earth?"

Firstly, I question whether the "scientific evidence" is all that scientific. Karl Popper said that a scientific statement is one that can, in principle, be falsified by observation. He may have exaggerated the role of falsification, but surely no one can deny that if a statement cannot possibly be tested by an observation, it is not in the realm of science. So the question, "Is it raining outside?" is scientific, whereas, "Is there a God?" is not.

Let us examine the statement, "The universe is billions of years old." Is it scientific? Are there observations that could confirm or not confirm it? I believe the answer is "no". To test it, we would have to construct hundreds (probably a much greater number) of parallel universes, evolve them in time over billions of years, and see where they wind up. We have no method of constructing parallel universes, so this procedure is impractical. That is, we have no basis for measuring the statement empirically.

Now let us examine the statement, "God created the universe a few thousand years ago." This statement fails to be scientific for nearly the same reasons as the other.

To sum up so far: no ultimate question about origins is scientific, because there can be no experiments to test it.

However, there is the creation account in the Bible, and as I take the inerrancy, infallibility, and inspiration of the Scriptures as axiomatic, I see no reason not to believe in that historical account. Science, as we have seen, has no tool at its disposal to prove or disprove anything with respect to origins.

There is another, mathematical, reason why science cannot disprove the idea of a mature creation: semigroup theory. Suppose you have a system that starts at state A, evolves in time through state B, and ends up at C. That is, the state evolves this way: A -> B -> C. Now, let's take an identical system, and suppose that it evolves in time from B -> C. Semigroup theory says that if you're in the system, and you have no outside knowledge or memory of the system, then there is no test you can perform to determine if you came from A or from B. So, if we take the universe as our system, and we have no outside knowledge or memory of its evolution in time, then we cannot tell at all whether the universe was created only a few thousand years ago to look billions of years old, or whether God really did create it billions of years ago, and its age and apparent age match. Therefore, science cannot disprove mature creation.

It has been brought up in comments, and in the edit to the OP, that the appearance of age requires deception on the part of God, and is therefore inadmissable. To that I would reply that we are no judge. First of all, our judgement of the age of the universe can be, and almost certainly is, spurious. We have no memory of those kinds of vast eons of time. Secondly, if we are talking about the Creator of the universe, then He is not answerable to us, but we to Him. Thirdly, I would point out various portions of the Scriptures that indicate that His ways are far above ours. If we do not have the full picture, and we do not, then we should not confuse our short-sightedness with clear-sightedness. Do we have the full picture? Do we know all the things going on here? I think not. If we serve a God Who is by no means obligated to reveal anything at all to us, we should not complain if a "few things" are unclear. Fourthly, God is Who He is, and we cannot change Him. Our conception of God is quite unimportant. What matters infinitely more is the conception of God as He has revealed Himself to us.

It has also been noted in the comments the various reasons why God might have created things with an appearance of age: utility, more glory for Himself. Somewhat related to the second reason there is my favorite reason: for beauty. The stars would surely be more beautiful if their light was already streaming to the earth.

Another alternative is the white hole cosmology proposed by Dr Russell Humphreys. This approach, fully supported in peer-reviewed literature, simply assumes that, instead of an infinite universe, the universe is finite. Then the equations of General Relativity produce a white hole, which can completely explain time differences and streaming starlight.

There are a few other aspects of this question that I have answered in different details on this blog post.