Is either cat unneutered? Why did you throw out the rugs? You can clean them with vinegar.

At the very least keep old rug around to use to make scratching posts.

Well, this sounds awfully familiar. I have had the similar problem and I have, atleast I think I have, explored every route possible (from removing the carpet and replacing it to letting the vet do a ultrasound) to discover the cause for why my male and neutered cat pees and defecates on the carpet and in only one room in the house. Usually vets say, lets check out to see if there are any biological reasons for why your cat is doing what he or she is doing. My response, dont try to rule out the biological stuff first because thats expensive. Yes, your cat is new so that could cause you to ask: is that why he is doing it? And that may have something to do with his reasoning, but is there a reason why he would like that bedroom? Perhaps it is the quietest room in the house? Your cat is telling you, (outside of a possible biological reason) that he also prefers a carpet textured area to take a pee and not the normal kitty litter. If your kitty is still pissing on the area where the rug was or in it's vicinity consider what I said about earlier about your bedroom being the most comfortable environment for your cat to piss. If he has moved to other floor textiles in your home, then perhaps it is specifically the carpet texture. But in either case, whether it is the room or the carpet, and he is still pissing put a thick (so the pee doesnt soak through it, think a few sheets) pile of newspaper down and see if he will piss on that. I swear out of all of the advice and tests done on my cat, it was the receptionist who told me about the aforementioned and by god she was right. So instead of going on the carpet, though he hasnt changed his room preference, my cat goes on the newspaper and I just throw the newspaper out after its soiled and put new sheets down immediately to cover the area.