They are all related to "regulate the air temperature", but they don't function the same:
air-conditioned: past participle (過去分詞)
air-conditioning: present participle (現在分詞)
An air-condtioned room is to use an air-conditioner, which is an air-conditioning system, to air-condtion the room.
I know the example sentence is kind of tongue-twister (繞口). I just want to show you how these words can be used together to demonstrate their individual functions.