For Gaia, it would be impossible (or too expensive) to put the whole catalog into a conventional DB. The solution is probably going to be something similar to OpenDAP (https://www.opendap.org/), but I haven't looked much into it myself.

Any plans on integrating with the Virtual Observatory, or is that too slow to integrate nicely?

One of the goal for the web version is to allow displaying sky observations yes (we are working on it). Not specifically from the VO, but it could as well.