removing the backplate is pretty simple mate, just undo the screws/nuts and bolts and lift it off. there shouldnt be any more to it.
some coolers have foam spacers that attach with gum to either the backplate or the motherboard. either way just peel it off and clean up any residue. (an eraser is good for this)
as to whether you can use the old 1, that will depend on the new cooler and how it attaches. some will attach directly to the pcb while others will need to pass though the mounting holes and attach to the backplate. if it passes rite through and attaches to the backplate then its likely that it provides additional bracing, if this is the case you will need the 1 that comes with the cooler as it will provide the rite depth spacing...