In the Web Vault, none of the saved preferences survive a browser restart — I believe that is expected behavior. There is a relevant feature request here:
On the other hand, I noticed that if you keep the browser open and log in to the Web Vault in a new browser tab (whether or not the tab where the Web Vault was previously logged in has remained open), then all Web Vault preferences are remembered — except for the “Show website icons” setting. This behavior seems like a bug to me.
For me, with Firefox ESR, all those options except the one mentioned by @iammike are kept across browser restarts. If I were to guess, I’d say that the option that doesn’t persist is a bug since it is the only anomaly.