We had 3 whole inches that shut the whole county down. My little guy (12 whole lbs) hates snow. It literally comes up to his belly.
So an option... (note to self during next hurricane)....
Anyways ... get a kiddie pool ... and put plastic down inside of it and top it with some sod ... put it out in the garage, patio or wherever .... that way your pet has a place to relieve themselves without fighting the elements. The sod can be dumped later. I think it would be smelly after awhile, but when you have extremely inclimate weather it sure beats them doing their business in the house or you having to go outside with them.