If everyone knows something, isn't that equivalent to there being "no possibility of someone acting as if the claim is true"?

"Any newly arrived cat at Anjellicle Cats Rescue will not be available for adoption if there are others cats waiting for longer than four weeks. Nittens is available for adoption."

We can simplify the first statement to "If a cat is new and there are other cats waiting for longer than four weeks, it will not be available for adoption."

The contrapositive is "If a cat is available for adoption, the cat is either not new or there are no other cats waiting for longer than four weeks."

Therefore Nittens is either not new or there are no other cats waiting for longer than four weeks.

Yes! A set can contain another set (look into powersets for an example of this). A set can contain only one entity - this is called a singleton set. A set can also be empty.