This doesn't answer your question about the lodging, but does about the false positive. If I were to get a positive test result that I was pretty sure was a false positive, I'd not submit that result to the airline. I'd get another test that would probably be a true negative. If that one was also positive, then it's a pretty sure bet that I am, in fact, infected. The CDC requires a negative test, and they don't have to know about the false positive.