The tile guy was trying to be just a little TOO good. You may need just a little bit more clearance to get the stem out. ( the one nut shown in your pic is the packing gland. you need to unscrew the whole body, with a larger socket further towards the inside)
I use a 1/2" cold steel...
The walls must have a waterproof substrate such as cement board, with a membrane under it.
The floors must have a waterproof membrane ( vinyl, lead, copper) with mortar preslope bed on top of that.
Lots of books available. Do some research before you tackle this. It is a lot of work...
There are many different types of thermostats, depending on what type of heat/cool system you have. You need to read the specs for the new one to determine if it is compatible with your system.
Kissler or Lordahl, or Danco for that matter, probably make a handle. You need to go to a large plumbing supply. The home stores only carry the very common stuff.
It appears from the pictures that this is a ceiling fixture which was put up with plastic expanding drywall anchors. This type of fastener is TOTALLY UNACCEPTABLE for a hanging load, even a relatively light fixture. These anchors have excellent strenghth in a wall with a vertical load...