"Why is God, rather than the universe, the only thing allowed to exist without a cause?"

He is not. If the universe were past eternal, then the universe would NOT require a cause. Sadly, for you, the evidence of the last 150 years (Big Bang, cosmic background radiation, 2nd Law of Thermo, positive inflation rate of the universe, and BGV Theorem) has gone away from an eternal universe and toward our universe having a beginning. (That is why Premise 2 is not being attacked as much by a-theists these days as it was 40-50 years ago in my youth.)

"Virtual particles pop in and out of existence all the time all around us."

No they do not! Actual particles, quarks and stuff, go into and out of existence due to the rearrangement of QM fields, which are NOT nothing. They have a cause.

"Even if the universe had a cause, why would it have to be God? What's your evidence?"

We know that since space, time, and matter had a beginning at the Big Bang, this Cause MUST transcend space, time, and matter. In other words, the Cause must be spaceless, timeless, and non-material. We also know that this Cause MUST be immensely powerful, right, in order to create 100 billion galaxies out of (literally) nothing?!? We can also surmise that this cause must be personal, in some sense, as It has chosen to create, and only personal agents can create, to our knowledge. Moreover, this Cause has chosen to create (or allow the creation of) persons (that's us!) - indicating strongly that It is personal.

This Cause is also self-existing, right? We know that either the universe (or multiverses, if they exist) are self-existing OR the Cause of same is self-existing. (Those are really the 2 options we have.) But, since the secular data points toward the universe having a beginning (and overwhelmingly so), then we must conclude that the First Uncaused Cause is self-existing. There is also a way to argue that this Cause is immutable or changeless. Let's not get into that too much, but it's worth thinking about on your own, OK?

So, we have: spaceless, timeless, non-material, immensely powerful, personal free will, self-existing, changeless. All we are really missing is omnispresent, omniscient, and holy. There are arguments there as well, particularly for the first two. But, we have Something that looks a LOT like the Judeo-Christian God.

"Moral values exist without God just fine."

We are talking about OBJECTIVE moral values and duties. No shifting the goalposts.

"The fine-tuning of the universe only appears so because we evolved to fit it."

Not true: the fine-tuning argument shows the improbability of life of any kind evolving in this universe, whether we evolved or not.

"Since the vast majority of our universe is uninhabitable to us, it's not very fine-tuned anyway."

Also false. Both galactic habitable zones and circumstellar habitable zones show that these are necessary for the existence of any life whatsoever just here on earth. The currently uninhabitable portions of the solar system, galaxies, and universe still feed the other portions - in terms of necessary conditions for life here on earth. Our moon, while currently uninhabited, stabilizes the tilt of our rotation. The Sun, while not habitable at all, is quite useful - it is the perfect star for us to be orbiting - in terms of size, type, power output and stability, and distance from us. Jupiter does an excellent job of sweeping asteroids out of our way and also of helping keep our planetary orbit more stable, as its gravitational effect on us is not inconsequential. Moving out into the galaxy, we need those heavy elements from supernovae, but, of course do not want to be too close to them when they occur. Our location in the outskirts of the Milky Way is perfect! Many galaxies are largely uninhabitable due to their star densities, but do a great job of feeding other galaxies with stellar material - absolutely essential to life here.