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Just my opinion: God can create anything at any age He chooses. That's why you can't take the carbon dating literally...He can make a 10,000 year old rock, or a 30,000 year old rock at the same instant....

And populate the universe with copious evidence that it's a very, very old place?

True, God could do that. God could have created a universe a mere 6,000-odd years ago, and created it indistinguishable, on day one, from a universe that was billions of years old.

But wouldn't that make God a liar, trying to ensnare man with false yet irrefutable evidence of a past which had never occurred?

I don't believe God can be a liar. I think that the convincing evidence he left pointing to a very, very old universe means that it is an ancient universe, and that a literal-minded reading of Genesis is wrong.

I've never yet read anything in ancient literature which seemed to be claiming that the entire universe was created recently. Creation stories I read invariably appear to be speaking of the creation of this planet and its immediate surroundings (our solar system) and often refer to the appearance of the Earth and heavens as they appear in changed versions after some calamity or large-scale event. The most obvious case occurs late in Isaiah:

ISA 65:17 For, behold, I create new heavens and a new earth: and the former shall not be remembered, nor come into mind.
ISA 66:22 For as the new heavens and the new earth, which I will make, shall remain before me, saith the LORD, so shall your seed and your name remain.