How can an in-floor electrical plug in box be installed in the middle of a room (such as a great room) when the home is already finished and the ceiling in the room below is sheet rocked?

You will have to run a snake down the wall that is perpendicular to the floor joists. Then, you run another snake from the hole in the floor, "catch" snake #1 and pull one snake through. Attach wire, pull to new floor box, and voilà!

Now all you have to do is connect it to a circuit that has the extra capacity you need. Otherwise, you'll have to snake all the way to the circuit panel, which could get REAL tricky, depending on how far you have to go.

Regards,
Stew