I drive the screws flush with the board. Make sure you drill is set on the drilling option so it doesn't limit the power (don't have it set on a number).
I wouldn't countersink, but if you did you should be able to just scrape off the excess that came out. Shouldn't need to unscrew first.
You did put the tub in first right? If not, you might have a problem with the tile because the tub has a flange.
This detailing is important. Two ways. Both ways have tub in first.
1) Hardibacker up top of tub flange. Silicone space in tub flange. Tile.
2) Make frame 1/2" small. Notch studs to get tub in. Hardibacker goes over tub flange keeping it a quarter inch off of tub with shims to prevent wicking. (This is my preferred method, but may be too late for you if already framed).
You can always build the studs out with quarter inch strips of material.