Yeah, as you suspected, the entire metal tab on the back and top of the TO-220 is electrically connected to the centre pin of the package. In this case, that's the MOSFET drain pin.
I personally wouldn't bother with copper under the tab on the PCB. You won't be able to get enough copper there to make any thermal resistance with the current layout you have. Unless, of course, you have a ground plane there, in which case you'll have to isolate the three tabs from the copper (and each other). You'd normally do that with a silicone or mica pad, which goes between the tab and the metal surface it's mounted to. The bonus of using the silcone pad is that it's slightly squishy, and you don't need to use any thermal compound.
Also, you'll need plastic bushings which isolate the screw that you'll need to use to hold the whole thing together. Depending on where you get the pads, they may come in a pack with the bushings.
As a side note...if you get the TO-220FP package, rather than the vanilla TO-220, the tab is encased in plastic and you don't need to worry about isolating it. The drawback is crappier thermal transfer.