Where should tile backer board go on a bathroom floor?
Typically, you would install the shower curb/pan first, then install your underlayment, concrete backer board in this case, up to it.

As to the floor, we usually extend the concrete board as far as we can under the wall finish, and have a gap between the bottom of the wall finish and the floor finish that gets covered up by the base to mitigate water damage if some water gets under the base.

When we get freaky about it, we paint and sometimes redguard the bottom of the base before installation, then clear silicone caulk the base to the floor after installation.  Mostly we do this if there are young kids that are going to be using the bathroom and the clients are ok with the mild funkyness of such an install.