[pushkins]

The threshold is the transition from the new tiled bathroom floor to outside the bathroom floor. You should be first installing a backer board on the bathroom floor first (screwed and glued), then your tiles, then the threshold then grout. (unless th threshold covers over the tile then threshold is last.
I cannot see why you would need to cut any hardwood floor before tiling, basically the backer board is laid over the floor. As a giude when the door is shut you shouldn't be able to see the new tile, so the exterior edge of the bathroom door is the line for tile/transition strip.