Walls – Will faced insulation behind poly sheeting trap moisture

I would probably use unmodified thin set to cover the gaps and give a good base for applying the tile.

Could you do something a little decorative and use a border tile around the repaired area, sort of like a picture frame? Then use a different tile to fill in the area. The border would make the change in depth harder to see.

Provided you don't have knob-and-tube wiring, this one is easy. Remove the vinyl siding, cut holes through the EPS foam and the sheathing at the top of each stud bay, and inject dense-packed cellulose into the empty stud bays. Should be pretty cheap and help a ton. I wouldn't use retrofit-style non-expanding foam. It'll be more expensive, highly flammable, and some people have allergic reactions to it once it's installed. Then add as much additional EPS foam over the existing stuff as you want. XPS foam, Polyiso foam, or rigid mineral wool will also do fine. Then install 3/4" thick vertical furring strips (usually 1x3s) over that exterior insulation to form a drainage channel (important since you're in a rainy climate), and then nail your new siding of choice to the furring strips.

If you do have knob-and-tube wiring, you need to keep the stud bays empty until you have it entirely replaced with modern cable.