Very poorly. The 27MHz radio used here won't penetrate more than a meter or so. GPS doesn't work through more than some inches. But like most things, the lower the frequency and higher the amplitude the farther the penetration. Acoustic modems work better under realistic constraints, though with a few exceptions you need the line of communication to be rather vertical if you're talking very long distances (hundreds or thousands of meters) because the speed of sound in water varies by pressure/temperature/salinity so the signal path curves up back toward the surface (from Snell's law).
Sonar, which of course works by sound, has issues because of the speed variability you mention. However, IIRC (and in salt water) the range of an active sonar at the right frequency (lower is better for range) is on the order of 20 thousand meters. That said, a submarine can hide from sonar under a thermocline, assuming the sonar is above the thermocline.
Absolutely could! Some do this. It depends on your chosen operational constraints. Keep in mind that neutrally buoyant cabling isn't particularly compact, so it creates its own set of problems.
> You can't go infinitely deep anyway, because eventually the pressure will penetrate the seals.
This is actually the least useful reason why you can't go infinitely deep. For exploration you will basically always want to be near the bottom of the body of water because mid column is just extremely boring, so you will want to make sure that your pressure tolerance allows for that. We just haven't found any infinitely deep bodies of water yet.
