General rule is 1 1/8 subfloor including the substrate, in your case the backerboard.
Yes you should cut the door casings to fit tile underneath, but not the jambs. A good way to do so is to use a special saw designed for doing exactly that called an undercut saw - you should find it at your local Home Improvement Chain Store for cheap. Just remember that when you cut, lay the saw not only on top of a tile for the proper height but also lay a small piece of cardboard down on the tile to represent the thickness of the thinset that will be there when you lay the tiles. The door clearence can easily be adjusted by cutting the door. Remember you can pop the door off its' hinges by removing the pins in the hinges and not have to weaken the hinge contact with the jamb by unscrewing it. Your buddy had a good suggestion, scoring the door first will definitely guard against splintering - i think you'll be surprised at how thin and flimsy your door really is when you cut it. When you do cut the door, if its the first time its been cut, there will be a solid bottom. When you cut it off, if you cut high enough that you get into the hollow part, you should chisel the masonite off of the solid bottom and reglue into the freshly cut door after you are done. Good Luck.