I'm planning on tiling the entryway around the cutout door to my
basement (about 3' x 6') and a hearth in front of a fireplace. (about
2' x 4'). Both are going to be ceramic tile, and are going to be
surrounded by wall to wall carpet on 3 sides. Right now, the floor is
an unfinished cement floor, and the carpeters are coming in a week or
two, so I need to get moving on the tile.

After talking to a guy at the store, we agree that I probably need a
backerboard under the tiles, so that the height is proportionate to the
surrounding carpet. (Moderately thick pile on top of a 3/8" moisture
resistant pad). So I bought some 1/4" Durock cement backer boards, and
was getting ready to start slicing them up and putting them down, when
I thought of some more questions.

1) Do I need to (i) tape the seams and/or (ii) put down some moisture
membrane on the area by the door, or can I just seal my grout and
figure that it's cement all the way down?

2) How should I attach the backer board to a concrete subfloor? Will
thinset be enough to hold it in place, or should I mess with masonry
bits and screws, and if I need to screw it down, what kind of bit and
screw should I use?

3) Is there a good way to figure out if I've got the tile at the right
height? Would I be happier with 0" of backerboard (i.e. none), or
1/2"?

Thanks, TM