Is it true that space had no volume at the Big Bang? I understand that density rises without (known?) bound as we approach the Big Bang, but even if you compress infinite space by an infinite factor you might still have infinite space -- infinity divided by infinity is undefined.

If you run the density backwards far enough, eventually everything in the visible universe is effectively at a single point, in some theories this was modeled as an actual infinite singularity, but it’s quite possibly unknowable since our first evidence is the CMB 380k years into the whole thing. As we rewind time from there things being to get highly speculative.

But if we're looking in the direction of a galaxy far far away (i.e. 14bn ly away) (which we don't know it's there, but let's assume it is, because likely some galaxies must be there), how come we see the CMB, which is 13.7bn years old, and not the galaxy itself, which is (by definition) less than 13.7bn years old?

The light we now perceive as CMB was produced everywhere, including our neighborhood; there was no place where you would not see CMB. Thus as the time progress, the bubble around us where that light has faded away is growing, while the light from far galaxies is only just approaching us.

Because what we see from that direction are the things that were there in the last 13.7bn years. But the farther we look we see older images. So while we can see image from 10bn years ago of something that's 10bn light years away, when we try to see something 13.7bn light years away we just see microwave soup because that's what was there 13.7bn years ago. And if we try to see something 14bn light years away we see nothing from there because 14bn years ago there was nothing there.

To "what exists today, beyond 13.7 billion light years", I'd rather say that "it doesn't exist yet". Because, otherwise, it assumes there's some "global universe time" like UTC or something. But all there is is our light cone.

That is fair. We are veering into metaphysics or religion if we talk about what exists there today, or even if there could be any relationship between our “today” and their “today”. By definition there is no causal connection. I’ll get back to you on that once I finish inventing my Time Machine.