I disconnect them because I store the scope in its bag so it doesn't rust in my area's weather.
As for the cat problem, get him a chew toy, an already broken thing he loves to chew on, and get it near the oscilloscope probes. Now every time the cat gets in chewy mood, just yell at him if he chews on the cord, but ignore him if he chews on the chew toy. Works fine for me

At least when I watch him

The cat usually enters chewy mood when I am around and when he's on purr mode, I don't know how yours behaves. I don't get broken cables anymore.