Yeah, Cairo will do the same to a blanket in there if he can't get the door open. He hardly ever goes in the baby cage more than a couple times a week, no more than ten minutes at a time.
RV/Caravan places carry a non slip item for cabinets and such that could work wonderfully for you. I have used a similar product used as a base for rugs on hardwood floors as well. It washes up easily, too.
Zahra and my Joey are very similarly marked. They almost look like twins! I too am fortunate that Joey doesn't climb on the counters or table. It's fun (at least for me 😃 ) to see one on the counter!! Thanks for sharing photos. I need to figure out how to get photos up.