Tile Cleaning – How to Remove Spray Paint from Floor Tile

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.

In the to-do section of the care instructions you posted, it suggests "Use an all-purpose, non-oil based household or commercial cleaner." OR, it says, you can use vinegar/water.

I think you are over-analyzing. The normal floor mopping solutions like Pine-Sol or a Swiffer will work fine. Just don't use anything abrasive that will scratch the tile or an aggressive bathroom "tile/grout cleaner" that is meant to bleach and renew grout that is stained.

If you're concerned about a particular cleaner, you could always test a small section that is out of sight. You can also apply a "grout sealer" solution over the grout lines. This is tedious, but can help slow down staining of the grout.