The negation of “The universe is seven thousand years old or less” is “It’s not the case that the universe is seven thousand years old or less.” Assuming the universe exists at all and time is real, the negative statement implies that the universe is more than seven thousand years old. So you’re just nitpicking there.
As I said, if you want to entertain radical skepticism, including the alternatives that the universe doesn’t exist or that time is unreal, you’re up against the pragmatist and ethical defense of science (e.g. the objection that hypocrisy is bad).
The domain of logically possible theories may be infinite, but scientific work isn’t purely logical or a priori. Scientific work is empirical, which means it’s based largely on observation and that narrows down the options. There aren’t an infinite number of theories that are both plausible to the human mind and that make sense of all the evidence indicating that the universe is billions of years old or that a human can’t fly by flapping his arms.
Yes, a theory is a large body of statements and predictions. As the holists such as Quine showed, some parts of the theory are more central than others. And as Kuhn said, there are social factors in switching from one theory or “paradigm” to another. Epicycles were added to the geocentric model before physicists switched to the heliocentric one. But any such social factors would be consistent with my pragmatic account of scientific knowledge.