What to do when removing Shower Fixtures from wall?

You don't need to remove the faucet unless you are replacing it, It has to be there when you put the backer board up, so does the shower riser and drop ear elbow. The only things you should remove are the shower arm where it threads into the drop ear elbow and the faucet handle(s) and escutcheon plate(s) You really don't need the water off to do this except when removing the old tile and board. Green board was fine as a shower backer it is likely you have a leak back there somewhere that caused the problem so look carefully before you put the backer board back up or you could be in the same boat again soon. It could be in the shower riser so look over that carefully. GL