Flooring – put snap together wood flooring over bathroom subfloor. Is this effective

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)

I think I would jack the wall up a tiny bit (1/16 inch), enough to slide out the particle board and slide in plywood.

Given the difficulties of jacks being in the way of the work, cut the plywood so it is only 12-16 inches wide (and 8 feet long or whatever) so the jacking support can be away from the wall.