Door trim: I assume you mean the bit of wood that goes around the door facing. That should be cut at the bottom to allow the tile to fit snuggly (not tightly, actually barely touching the wood when set). The Frame is the bit of wood that goes around the door itself. The stop is the narrow piece which actually stops the door from swinging the other way. The reality is that all of these bits can be cut to accommodate tile. Its not too hard to get a nice straight cut if you set a tile or a couple of tiles next to the wood bit and use a hand saw resting on the tile while cutting. This will give you just about the right depth of cut. After you tile and grout and seal the tile (if sealant is needed) you will go back to the door with chalking - preferably a paintable type. You fill in the small crack that will be there, using your thumb to finish it off and to "press" the chalk under the trim/frame/jab then touch up with paint to match the door. How it would look would depend on what the tile is (stone, ceramic) and the color/texture. Mind you will need a bit of molding along the tile to floor side. that transition molding or threshold molding will finish off the edge of the tile and will provide a smoother transition between the two floors. The step up will be about 1/2 inch - given 1/4 inch for the tile and the other 1/4 inch for the shoe molding that goes over the tile and the cement beneath the tiles. This is not too high of a step up. However it can play a tad bit of havoc for wheel chair accessibility, or other wheeled devices.