for the first question, send your pages so i can see how they are configured.
for the second question, the trigger you are using for setting the session is page load. This wikll set the session regardless of whether the searchbox form has been posted or not.
when you submit the search, it sets the session because the searchbox form element has a value, when you come back to the page, it sets the session to be blank because the searchbox form has not been submitted.
for the trigger, click the lightning bolt icon and select the searchbox form element same as the value so the session will only get set if the searchbox form element is posted.