Under the disgusting carpet in my living room, I've got hardwood floors.
Since part of the room serves as the "lose the shoes" zone, I'm tired of
trying to keep the carpet clean. So, I'm thinking of installing some sort of
tile or slate in that area - something easy to just mop clean. That "zone"
will meet new carpet at some point. Two questions:

1) Whatever backing belongs under tile, does it have to be so thick that the
area will be slightly higher than the adjacent carpet? I'm wondering if
that'll cause height difference problems where the two materials meet, and
people will trip.

Why not just carpet (or not) the entire room, and put
some sort of removable mat in the doorway area?
Then you can take the mat out to the porch area
to clean it.