Do all things exist? Nonexistence is the absence of existence, by definition. So, nonexistence does not exist. Therefore there is no such thing as nonexistence. To say that something does not exist thus seems to be a fallacy, since NOTHING does not exist. Everything, therefore, must exist. Is this right? If not, what is wrong with the argument?

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Of course, in a perfectly good sense of "exist", existence doesn't exist either. Existence isn't a thing, and so there is no such thing as existence, though of course, bears, bells and BMWs exist, to mention but a few.

And yes: there is no such thing as non-existence, because "non-existence" isn't way of referring to a thing. But unicorns don't exist. Neither do square circles. And, according to some, neither do free lunches. No fallacy there.

Does everything exist? Well, if "everything" means "all the things that exist," then everything exists. (Though of course, this doesn't mean that there is a special thing, namely everything, that exists.) But since, as noted, unicorns don't exist, it's not true that "everything" in the sense of "everything that might have existed" actually exists. It's likewise not true that that every description (e.g., "round square") picks out something that exists. The conclusion of the argument comes partly from trading on ambiguity.

Related: 'existence' and 'nonexistence' are nouns. But that doesn't mean that they're meant to refer to things of a certain kind. Compare: I'm writing this post for the sake of answering your question. 'Sake' is a noun. But there aren't any sakes, even though I really am doing what I'm doing for the sake of what I said I'm doing it for. We use 'existence,' 'exists,' 'non-existence' and 'doesn't exist' to talk about what is and what isn't. Since there clearly are bells, bears, etc. and clearly aren't unicorns and square circles, we can equally clearly say that bears, etc. exist and that unicorns, etc. don't. Notice, however, that so far, we've used the copulative verb "exists" rather than the noun "existence." We can also say things like "the existence of bears is a fact about the world, as is the non-existence of unicorns." But it would be bad business to take the surface grammar here as a guide to metaphysics and to infer the existence of EXISTENCE and NON-EXISTENCE. We get ourselves into philosophical hot water when we turn accidents of language into ontological principles, and we bring the water to a boil when we slip back and forth between different meanings terms without paying attention to where we slide.

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