Drywall – What are small plywood blocks between flooring and drywall edge

Would a sill extension work for filling the gap at the entry door? Here's the first link I found: http://www.thebuilderssupply.com/Sill-Extensions_c_351.html

And, since you mention that its appearance is secondary, I'll throw out that you might be OK just filling that 1/2" gap with grout. On my first tile job I didn't notice that the edge of my fiberglass tub had a slight belly, and ended up with a gap that went from 1/4" at the middle of the tub (same as my grout spacing), to about 1/2" at the head and foot corners. I filled the gap with grout and it looked surprisingly good. When we moved after about 4 years there was no evidence of any cracking or separation between the tub and the grout; it looked pretty much the same as when I first installed it.

EDIT: Another possibility would be to fill the gap with a length of aluminum bar stock to match the threshold. You could remove the threshold and fasten the bar to its tile-facing edge with countersunk flat head machine screws and nylon-insert locknuts so that the screw heads are hidden in between the tile and the bar. Assuming the cement board is butted right up to the threshold, you'd probably need a 3/8" square bar mounted high on the threshold so it overlaps the cement board.

I think you should be more concerned with the expansion gap than other types of expansion.

In my opinion you should use some kind of separation between the living room and the bedrooms, because bigger rooms require bigger expansion gaps. And that difference in expansion could give some bowing.