The engine bay does get fairly hot (your car like mine is black). When I go to wash the car I have to be carefull around the scoop cos the heat rising up through the intercooler then out the scoop dries the water/carwash pretty quick (matter of seconds) if the car had been driven say more than around the block
I confirm my O8 tuned by STi wagon does not have a build number inside the car. I bought mine brand new this year in Feb, still only just reached 5,300km
I havent checked since my last service they havent nicked my engine cover and put a standard one on it... only reason they might do it is to do some cosmetic mods to another car then try sell it as a limited edition to make more money off an unsuspecting doofus. My dealer doesnt seem as shonky as some i read about over east, so id be extremely surprised if my cover was missing.
Good luck but i dunno what solution they can offer regarding missing build number. They havent unstuck it then glued it onto the likes of your radiator support rail like HSV's which is better so if owners do engine mods and if they cant refit cover they still have their badges?

