In my experience the best tool to use are sharp scissors. After you have cut the healthy pieces off, you should take the rock out of water and carefully remove the dead sponge from the rock.
Again, IME, for sponges the best way to secure them in place is to just push them to holes in rocks or between the rocks while trying to apply as little pressure on to the sponge as possible. If this is not possible, you can use plastic toothpicks or fishing line to secure them.