In the Castaway's Cove I am unable to move or reposition a rug once it's been put down. Certainly this isn't a game-breaking issue for me (and I'm enjoying the game immensely), but it's annoying to have to pick up all my items and reset everything in the house to move a rug.
I noticed this as well - any rug, once placed, becomes impossible to select from the housing placement interface. I *think* I found a workaround, maybe. Open the Attic, and switch to the House tab. This shows all the currently placed items in this part of the house. I think it might be possible to pick up rugs from here. The downside being that if you have more than one of the same kind of rug, it's hard to tell which is which. And if any items are sitting on the rug, those will be picked up as well. Which makes it kind of... not good if you just wanted to move a carpet a few slots to one side, or rotate it.