Your rug is AT the back of the room. It is situated at the back of the room. It is not IN the back of the room, it is AT the back of the room. Something 'in the background' is there but is not noticed. I.e. you are looking at your friend, but there is a painting or window behind her, so those things are in the background, but your focus/ your attention is to your friend.

Back of the room. We use background to refer to somewhere nondescriptive. "The rug is in the background." Its like saying "it's there but it's not important to tell you precisely where" Background noise vs. The distant clatter of a train Background music vs the name and performer of the song But no we wouldn't describe a rug as background in the context you mention at all