@Mack

at least it seems reasonable to believe the universe does exist, though some might disagree

Things obviously exist, even if this universe is a simulation or something, so still the problem remains.

I don't think paradoxes (at least of this sort) can actually exist in reality

It's funny you say that, because I think that literally everything is a paradox and every truth comes with it's polar opposite.

Many people believe in some sort of un-caused first cause, like a God

That makes no sense, especially as an argument for God. If uncaused things can just pop out of nothing it could be anything, who's to say it would happen to be a sentient and omnipotent spirit creature or not?

I never know when I should use a capital 'G,' but certain members of this site will freak if I don't

Technically God should be capitalized because it is both a name and a title.

This would be a supernatural being, so they wouldn't necessarily be bound by natural laws like cause and effect

Something existing without being caused is like something having weight without having substance. But apparently something existed without a cause, and that something is the basis and substance of all that exists. In fact it is still creating things out of nothing, like the virtual quantum foam particles that pop in and out of existence.

which I think seems to be a natural law rather than a logical law

The way I see it nature (i.e reality) is what defines what is "logical". And mother nature is telling me that she doesn't make any bloody sense and that true logic is the opposite of logic. Logic is illogical.

Physics around the Big Bang gets really weird

It was infinitely small and infinitely large because the singularity was everything yet it was nothing, it was the infinitely dense center of nothing and the whole nothing and everything was embodied by a single cell inside of itself that formed the beach of the ocean of paradoxes as one spec of sand that multiplied itself infinitely.

we simply can't trust our intuition about the nature of cause and effect and time and all those things when it comes to that. For example I don't think it makes sense to say "before existence." Given the weirdness it makes sense to withhold judgment, even if you're an expert in the subject.

I agree about cause and effect, that is why it is a paradox. It doesn't just not make sense to us, reality itself doesn't make sense even by it's own rules. Thus logic is illogical and the greatest truth is also the biggest lie. We have a fabric underlying everything we see made of microscopic paradox balls (particles) that pop in and out of existence randomly and only take specific trajectories when they are observed.