If your pet, without your prompting, uses its claws to cut toilet paper on Shabat, you can use the cut toilet paper. (I actually asked this shilah and that was the response I got!)
I am not sure about this. If the toilet paper was muktzah bein hashemashos, it remains muktzah the whole day. Really, a Jew can also tear the muktzah toilet paper with a shinui, in a case of dire need, but if something else (separated tissues, napkins) is available for cleaning, or a goy is available to tear the paper, then…