OK, I'll continue with no regards to the framing at all.
I'd make sure all the tiles are stuck well by taping them with the handle of a screwdriver to listen for hollow sounds. Especially around the perimeter. Even tiles that are loose will still lay there nice and pretty and no one will notice.
Clean and rinse as you described. It's best at this stage to scruff the glazed to create bit. (I've done it without doing this, but you really should.) At minimum run a belt sander with the most coarse paper you can get, or better yet, a diamond cup on a side grinder or similar. This can get dusty so have help with a good vac. The rise etc. of course.
Use a good quality modified thinset per directions. But Mapei makes a primer for just this purpose, be safe, find some. Eco Prim Grip.http://www.mapei.com/us-en/products-line.asp?IDMacroLinea=0&IDTipo=6737&IDLinea=102
Plan ahead how you're gonna finish doorways. Be sure to leave the proper expansion space at the perimeter and all solid objects. Be sure to include expansion joints within the field at strategic points and every 20-24 ft. +-.
There's many other thing, but can't write a book here.