I am about to replace some tile and some carpet in a bathroom with tile throughout. I have two issues that I know that I will need to address. (1) The section where the carpet will be replaced butts up against some engineered hardwood, so I have a slight variance in thickness between the two floor coverings. If I use cement backerboard over the 15 year old concrete slab and under the tile, I will be almost exactly the same height for both floor coverings. Now the first question. Can I attach the concrete backerboard to the floor using thinset or some other form of mortar, or will I have to drill the slab and attach the backerboard with concrete screws?

Second issue. When I remove the old tile, I am almost sure that I will have residual thinset from the old tile. In the past, I have used a cold chisle and spent hours cleaning the thinset down to the slab. Because I need to also raise the level of the floor to match the backerboard, can I use thickset mortar (consistency of sand)to build up and level the floor where the thinset residue from the old tile exists; and then tile directly over the thickset mortar when it dries. It would be like doing a sloped pan in a shower.

Ok, I thinks you going to pull up the carpet and install ceramic tile. Well engineered hardwood is 3/8" thick so put tile next to wood also keep in mine there is the thing set will be under tile. read the beg it gives you trowl sizes for size of tile you going to use hight of that add to tile hight. that is your hight. Also with hardwood floors must not butt up to any thing. leave space in between. need to use trims made for this. It's called T- molding if you end up close to wood but if you are realy low from the wood than need reducer, I hope You don't go the way with board under it #1 not usual thing to do over concrete floor if so there is floor levelers out there will do same job and much faster. then screwing to cement and glue it down. drill bit and screws alone is arm and a leg. You second issue. pulling up the tile there couple of ways to go at this one way back braking labor use a bar they sell this at Home Depo to brake of the tile from the floor hammer tile first loosens it up or rent a hammer chisle let the machine do the work. Just make sure all the loose and high stuff is off the floor sweep real good. If you want to make more work for you self, rent a cement grinder and go at it.sounds like you like doing that.
Note: Do Not cheap out on the thing set use Flex I, Flex II or Full Flex. Also if tile ends up higher than towlet drain make sure to double up on the wax ring to get a good seal. Well good luck with you project.