If you are going to be covering the spigot so that it is never used, you might just want to remove and cap that part of the pipe (leaving the feeder to your outdoor kitchen). Spigots are known to leak from time to time, and having that happen once this is buried in your desk would be a pain in the butt to fix. As for hiding the pipe that runs along the floor, it doesn't look like it is too far from the wall. Could you get a thicker piece of material to use as a baseboard, and route/rabbet out a path along the backside for the pipe? Then put it in place and it would just look like trim, with the pipe concealed inside/behind it. Added benefit: if you ever need to get to that water line, you only need to remove a piece of baseboard.
glue and weight to hold it down while glue dries is all that is need imo, the baseboard idea is good also for sure and will work.... maybe a 2x base that can be routed out in the back, just won't be able to notch out the backside to accommodate the stonework as easily.