Tiles or Hardwood: which one better handles movement across a subfloor joint?
I will answer one of your questions:
Put a transition in.  Given the opening, I would put a single strip of tile that matches the frame depth,  that contrasts with the tile on either side of it.   So if you go with light coloured tile in both old and new part of the house, put a darker tile as the strip.
If you want to go with wood, do a single, or a few strips crossways along the joint.  Run the others against that.
Put a tile line exactly on the junction of old and new. This way if part of the house shifts, the crack will be in the grout instead of breaking a row of tile.  This may mean a double row of smaller tile.
Having a visible marker at the transition will reduce tripping if you can't level out the difference between the two sections of the house.
You probably should do a layer of Ditra or other isolation membrane for a project this big.  I have twice not used it, but my largest project was an 8x12 laundry room.
Trying to make a floor that flows through multiple rooms has issues:

If one part of the house moves relative to the other the flooring is under severe stress.  Having a natural movement joint relieves this stress,

Some years from now some accident (flood, fire, dishwasher leak, paint spill, uncaught dog pee...) makes it desireable to replace the floor in one room.  Replacing a room in a much smaller disruption than replacing the entire level.