You only need PT lumber when in direct contact with the concrete. Putting the vapor barrier directly on the concrete may cause moisture/mold between the concrete/barrier.

Have no idea what bickered board is. Backer board?The concrete kind?Read the followinglink of wood floor construction for tile

There are better choices for underlayment for floor tile, like ditra. That will help equalize the two substrates.

The difficult part will be getting the new surface equal height to the old.

Pressure treated wood, because of its high moisture content, shrinks considerably as it dries. Even non-treated construction grade lumber shrinks, although less so. I'd probably let the lumber dry for some period of time before using it. In any case, I would not tile across the joint but instead treat it like a control/expansion joint in a concrete slab. Maybe, maybe if you used one of the anti-facture membranes??

How do you feel about carpeting? ;-)

Would you have the same concern about 'hard wood flooring' being installed across the seem of different surfaces (wood and slab)?

My biggest concern is that the height of sunken area is going to change throughout the year as the humidity changes. How much, I have no idea although you might consider looking at expansion coefficients for whatever type of framing lumber you are going to use.

Certainly wood would be a better surface than tile because wood has some flex. I'm wondering if some kind of engineered floating floor might not be the best idea.

I'm also wondering if filling the sunken area with concrete might make more sense? Then you would have a uniform substrate.

Buy seasoned used lumber for joists and have tile installer bridge between different floors with urathane caulk. Tape is applied bridging gap then urathane applied over tape. The idea is tape come's loose to slide to and fro while urathane adheare's only to sides of tile and stretche's x contract's with movements. Urathane come's in colors close enough to basic tile colors. To view example of urathane,look for parking garage expansion joints,filled gap between commercial buildings and sidewalk,brick/stone wall expansion joints and other masonary work.

Even if the low room were filled with concrete the joint will have uneven movement and never be fit for normal tile. Think about required expansion joints in concrete mentioned above.

Many wood floors snap togeather and float,have a look at those.