Is it OK to use 1/4" concrete backer board over 1/2" plywood as a bathroom subfloor?
1/2" plywood is for temporary applications, exterior sheathing and used as gigantic shims. It does not belong as a floor or a roof; minimum 5/8" exterior grade.

HardieBacker:


  What subfloor should HardieBacker board be installed over?
  The minimum subfloor specified is 5/8" exterior grade plywood or 23/32'' exterior grade OSB subflooring. HardieBacker board must be installed in accordance with local building codes and the floor must be engineered not to exceed the L/360 deflection criteria, including live and dead design loads, for the specific joist spacing used. HardieBacker board is not designed for use over concrete.


Durock:


  Subfloor should be minimum 5/8" exterior-grade OSB or plywood.


Wonderboard:


  For the subfloor, securely glue and fasten minimum 5/8" (16 mm) exterior grade plywood or OSB panels (PRP-108) to the floor joists. Floor joists should be spaced a maximum of 16" (40 cm) o.c.; a 3/4" (19 mm) Exterior Grade Plywood or OSB subfloor with I-joists spaced a maximum of 19.2" (48 cm) o.c. is also acceptable.




For the threshold you don't want something that overlaps, like this:



For a Hardwood transition, you want the type that abuts the tile, like this one-sided beveled threshold: (I've had luck filling against them with grout; complementary colored caulk is also available for tile-to-wood transitions. Standard thresholds can be ripped down, length-wise, to remove one bevel and then the bottom sanded down for a custom fit. Endeavor for absolutely perfect cuts along the doorway.)



Since you are transitioning to Carpet, I'd use something like this or this: (an abrupt leading edge is more of a concern then a reasonably high threshold)