Please see \"Latest update\" below - Niel S has found a local storage solution for the VIEW MODE part of this question - but I still need to find a solution to
You don't have any code to load the localstorage value and apply the class to listings.
I've updated your fiddle: https://jsfiddle.net/n2o70xgg/2/
This is what I added:
if (localStorage.getItem('viewmode') == 'alt') { $('.listings').addClass('alt'); }