Will isolating a clothes dryer make it safer if fire occurs

You can install cement board on the subfloor and then any flooring you'd like on top of that. The cement board isn't that thick and is typically used to firm up the floor before installing tile, but you could put whatever you like on top.

However, rather than making the place only able to contain the fire, I'd consider installing sprinkler heads in the room to put out any fires before they have a chance to spread.

Prefabricated chimneys have specific firestop spacers that must be used to maintain the terms of their listing, though they are often just sheet metal with a hole cut in it.

For constructed chimneys such as those like yours of masonry units, use firestop foam to fill the gap between framing and the masonry at every floor and ceiling level. This stuff is made by many of the same companies that make other materials used in construction. It's typically much like expanding polyurethane foam used as a sealant to fill gaps in construction, except it contains chemicals so the material will not support a flame, the stuff just chars. Regular foam will support a flame and puts off noxious fumes when burning.