Wiring – Code compliance of running EMT on exterior of exposed chimney inside

You didn't mention how tall the chimney is. 2 story house or single. Is there or was there ever a displacement for a fireplace, one flue or two? Lined or bare brick interior?

In general, the procedure is brick by brick. A small mini jack/impact hammer will speed up the process, but it can be done with a hammer and cold chisel. Most chimneys are independently supported and not structural to walls or the roof. there may be ties to framing members that you will have to cut or remove as you go through the roof or floor. You definitely want to use basic safety equipment like a dust mask, gloves and good eye protection. You can cut down on the dust and debris spreading in the house by hanging plastic from the ceilings around the work area. Use some painter's tape to hang lightweight plastic from the ceiling creating an enclosed area around your work area.

One cool trick on chimneys with a large flue opening and a clean out on the bottom is to drop the bricks one at a time to the bottom and have a helper remove them and stack them at the bottom. This is a bit neater and saves bringing bricks through the house.

If there is a straight open space accessible from above, you might consider polyisocyanurate foam insulation. A 2 inch board has R-13 insulation.

You might want to consider a wireless thermostat. There are numerous types (both proprietary and wifi) that would allow you to move the sensor elswhere even if the base station was on the heated wall.