Instead of using cement board behind the stove as you appear to be doing, you probably want to use drywall instead. If you can get fire rated drywall in the 5/8 inch thickness, that'd be fine, or else use two layers of 3/8 or two layers of 1/2 inch drywall. Fire resistant drywall has tiny glass fibers that hold the gypsum core together even when the paper burns away so that the gypsum core remains in place during a fire.
The reason why drywall has such a high fire resistance rating is because the gypsum in it's core actually has a lot of water (H2O) bound up inside it. The gypsum in the core of drywall is calcium sulfate dihydrate, with a chemical formula of CaSO4-2(H20).
When drywall is exposed to fire, the heat applied to the gypsum goes into driving that bound up water into water vapour. It's kinda like putting a pot of water on the stove to boil. The hotter the stove the faster the water will boil, but the boiling water won't exceed 212 deg. F in temperature no matter how hot the stove. Similarily, no matter how hot the fire, the drywall screwed to wood studs won't get any hotter than 212 deg. F until all of the water in the calcium sulfate has been driven off into steam. It's only once the bound up H2O molecules in the gypsum are all driven off as steam that the remaining calcium sulphate powder won't protect the wooden wall studs from the heat, and the temperature of the wood will exceed it's kindling temperature and start to burn. Thus drywall on top of wood studs provides a measure of fire protection because it allows time for the occupants of a building to escape the building before the fire spreads to the wall studs.
So, drywall makes a good fire barrier because in the event of a fire, the gypsum core of the drywall acts very much like a pot of water. No matter how hot the fire, the heat produced will go into driving the bound up H2O in the gypsum into water vapour, and the end result will be that the wood studs in the wall will never get hotter than the boiling point of water (212 deg. F). That will take some time, and that time can be used to evacuate the building. Once all the gypsum core of the drywall is entirely reduced to calcium sulfate, the wood studs will no longer be protected from the heate, and will start to burn. Thus, it is the bound up H2O inside the gypsum core of drywall that provides the fire protection rating you need around a hot object like a wood stove.
My understanding is that "fire rated" drywall differs from ordinary drywall in that:
a) it is 5/8 inch thick as opposed to 1/2 inch thick
b) it's core contains glass fibers that hold the core together and prevent it from falling apart during a fire.
If the gypsum core falls way during a fire, then there is no protection to the studs behind that missing gypsum core, and they will start burning as soon as the temperature of the wood exceeds it's kindling temperature.
So far as the floor goes, any non-combustible floor would work. I think a ceramic tile floor under and around your wood stove would be fine because it wouldn't burn (like carpet would).