Hi
I can answer (I hope correctly) some of the questions.
1. Your site is password protected, so I can’t see that link (you can reply as a private reply with credentials). What Display method are you using and does the method you chose ask for the results container and pagination class (or something like that) and did you set them or check to make sure that the default values were correct for your theme (inspect the HTML of the page you gave to see)? What theme are you using?
2. Not an easy way. The only potential way to do this would be to have tracking cookies and then somehow use that data. Not easy and not doable by me/us 🙁
3. At this time, you cannot, as the search cannot work without them. I do not know if another way can be found for the future, but for now that is how it works.